Simplify The Expression:${ \frac{\sin 7a^{\circ} \cdot \tan(-315^{\circ}) \cdot \cos 300^{\circ}}{\sin 150^{\circ} \cdot \cos(-75^{\circ})} }$

Introduction

Trigonometric identities are a fundamental concept in mathematics, and they play a crucial role in simplifying complex expressions involving trigonometric functions. In this article, we will focus on simplifying the given expression using various trigonometric identities. The expression involves sine, tangent, and cosine functions, and we will use these identities to simplify it step by step.

Understanding the Given Expression

The given expression is:

$$\frac{\sin 7a^{\circ} \cdot \tan(-315^{\circ}) \cdot \cos 300^{\circ}}{\sin 150^{\circ} \cdot \cos(-75^{\circ})}$$

To simplify this expression, we need to use various trigonometric identities, including the sum and difference formulas, the product-to-sum formulas, and the Pythagorean identities.

Using the Sum and Difference Formulas

The sum and difference formulas for sine and cosine functions are:

$$\sin(A+B) = \sin A \cos B + \cos A \sin B$$

$$\sin(A-B) = \sin A \cos B - \cos A \sin B$$

$$\cos(A+B) = \cos A \cos B - \sin A \sin B$$

$$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

We can use these formulas to simplify the given expression.

Simplifying the Expression Using the Sum and Difference Formulas

Using the sum and difference formulas, we can simplify the expression as follows:

$$\sin 7a^{\circ} \cdot \tan(-315^{\circ}) \cdot \cos 300^{\circ}$$

$$= \sin 7a^{\circ} \cdot \frac{\sin(-315^{\circ})}{\cos(-315^{\circ})} \cdot \cos 300^{\circ}$$

$$= \sin 7a^{\circ} \cdot \frac{-\sin 315^{\circ}}{\cos 315^{\circ}} \cdot \cos 300^{\circ}$$

$$= -\sin 7a^{\circ} \cdot \frac{\sin 45^{\circ}}{\cos 45^{\circ}} \cdot \cos 300^{\circ}$$

$$= -\sin 7a^{\circ} \cdot \tan 45^{\circ} \cdot \cos 300^{\circ}$$

$$= -\sin 7a^{\circ} \cdot 1 \cdot \cos 300^{\circ}$$

$$= -\sin 7a^{\circ} \cdot \cos 300^{\circ}$$

Using the Product-to-Sum Formulas

The product-to-sum formulas for sine and cosine functions are:

$$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$$

$$\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$$

We can use these formulas to simplify the expression.

Simplifying the Expression Using the Product-to-Sum Formulas

Using the product-to-sum formulas, we can simplify the expression as follows:

$$-\sin 7a^{\circ} \cdot \cos 300^{\circ}$$

$$= -\frac{1}{2}[\sin(7a^{\circ}+300^{\circ}) + \sin(7a^{\circ}-300^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(-193^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(193^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(180^{\circ}+13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

Using the Pythagorean Identities

The Pythagorean identities for sine and cosine functions are:

$$\sin^2 A + \cos^2 A = 1$$

$$\tan^2 A + 1 = \sec^2 A$$

We can use these identities to simplify the expression.

Simplifying the Expression Using the Pythagorean Identities

Using the Pythagorean identities, we can simplify the expression as follows:

$$-\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(180^{\circ}+13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

$$= -\frac{1}{2}[\sin(307^{\circ}) + \sin(13^{\circ})]$$

= -\<br/> # Simplify the Expression: A Comprehensive Guide to Trigonometric Identities - Q&A

Introduction

In our previous article, we discussed how to simplify the given expression using various trigonometric identities. In this article, we will provide a Q&A section to help you better understand the concepts and formulas used in simplifying the expression.

Q: What are the main trigonometric identities used in simplifying the expression?

A: The main trigonometric identities used in simplifying the expression are the sum and difference formulas, the product-to-sum formulas, and the Pythagorean identities.

Q: What is the sum and difference formula for sine and cosine functions?

A: The sum and difference formulas for sine and cosine functions are:

$$\sin(A+B) = \sin A \cos B + \cos A \sin B </span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mi>A</mi><mi>cos</mi><mo>⁡</mo><mi>B</mi><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>A</mi><mi>sin</mi><mo>⁡</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\sin(A-B) = \sin A \cos B - \cos A \sin B </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>A</mi><mi>cos</mi><mo>⁡</mo><mi>B</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>A</mi><mi>sin</mi><mo>⁡</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\cos(A+B) = \cos A \cos B - \sin A \sin B </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>A</mi><mi>cos</mi><mo>⁡</mo><mi>B</mi><mo>+</mo><mi>sin</mi><mo>⁡</mo><mi>A</mi><mi>sin</mi><mo>⁡</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\cos(A-B) = \cos A \cos B + \sin A \sin B </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span></p> <h2>Q: How do we use the sum and difference formulas to simplify the expression?</h2> <p>A: We can use the sum and difference formulas to simplify the expression by substituting the values of A and B into the formulas and simplifying the resulting expressions.</p> <h2>Q: What is the product-to-sum formula for sine and cosine functions?</h2> <p>A: The product-to-sum formula for sine and cosine functions is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>A</mi><mi>cos</mi><mo>⁡</mo><mi>B</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">[</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">[</span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)]</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>A</mi><mi>cos</mi><mo>⁡</mo><mi>B</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">[</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">[</span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)]</span></span></span></span></span></p> <h2>Q: How do we use the product-to-sum formulas to simplify the expression?</h2> <p>A: We can use the product-to-sum formulas to simplify the expression by substituting the values of A and B into the formulas and simplifying the resulting expressions.</p> <h2>Q: What is the Pythagorean identity for sine and cosine functions?</h2> <p>A: The Pythagorean identity for sine and cosine functions is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>A</mi><mo>+</mo><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>A</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sin^2 A + \cos^2 A = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9552em;vertical-align:-0.0833em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow><mi>tan</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>A</mi><mo>+</mo><mn>1</mn><mo>=</mo><msup><mrow><mi>sec</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\tan^2 A + 1 = \sec^2 A </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mop"><span class="mop">tan</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mop"><span class="mop">sec</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span></span></span></span></span></p> <h2>Q: How do we use the Pythagorean identities to simplify the expression?</h2> <p>A: We can use the Pythagorean identities to simplify the expression by substituting the values of A into the formulas and simplifying the resulting expressions.</p> <h2>Q: What are some common mistakes to avoid when simplifying trigonometric expressions?</h2> <p>A: Some common mistakes to avoid when simplifying trigonometric expressions include:</p> <ul> <li>Not using the correct formulas or identities</li> <li>Not simplifying the expression correctly</li> <li>Not checking the final answer for accuracy</li> </ul> <h2>Q: How can I practice simplifying trigonometric expressions?</h2> <p>A: You can practice simplifying trigonometric expressions by:</p> <ul> <li>Working through example problems</li> <li>Using online resources or calculators to check your work</li> <li>Practicing with different types of trigonometric expressions</li> </ul> <h2>Conclusion</h2> <p>Simplifying trigonometric expressions can be a challenging task, but with the right formulas and identities, it can be done. By understanding the sum and difference formulas, the product-to-sum formulas, and the Pythagorean identities, you can simplify even the most complex trigonometric expressions. Remember to practice regularly and check your work for accuracy to become proficient in simplifying trigonometric expressions.</p> <h2>Additional Resources</h2> <ul> <li>Trigonometric Identities: A Comprehensive Guide</li> <li>Simplifying Trigonometric Expressions: A Step-by-Step Guide</li> <li>Trigonometric Formulas and Identities: A Cheat Sheet</li> </ul> <h2>Final Thoughts</h2> <p>Simplifying trigonometric expressions is an essential skill for anyone working with trigonometry. By mastering the formulas and identities, you can simplify even the most complex expressions and become proficient in trigonometry. Remember to practice regularly and check your work for accuracy to become a master of simplifying trigonometric expressions.</p>$$