Simplify WITHOUT A Calculator:${ \frac{\cos \left(90^{\circ}+x\right) \cdot \sin \left(180^{\circ}+x\right)}{\tan 225^{\circ} - \cos^2(-x)} }$

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on simplifying a complex trigonometric expression without the use of a calculator. We will break down the expression into smaller components, apply trigonometric identities, and manipulate the terms to arrive at a simplified form.

The Given Expression

The given expression is:

cos⁡(90∘+x)⋅sin⁡(180∘+x)tan⁡225∘−cos⁡2(−x)\frac{\cos \left(90^{\circ}+x\right) \cdot \sin \left(180^{\circ}+x\right)}{\tan 225^{\circ} - \cos^2(-x)}

This expression involves various trigonometric functions, including cosine, sine, and tangent. We will simplify each component separately and then combine them to arrive at the final result.

Simplifying the Numerator

The numerator of the expression is:

cos⁡(90∘+x)⋅sin⁡(180∘+x)\cos \left(90^{\circ}+x\right) \cdot \sin \left(180^{\circ}+x\right)

We can simplify this expression by using the trigonometric identity:

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A+B) = \cos A \cos B - \sin A \sin B

Applying this identity to the first term, we get:

cos⁡(90∘+x)=cos⁡90∘cos⁡x−sin⁡90∘sin⁡x\cos \left(90^{\circ}+x\right) = \cos 90^{\circ} \cos x - \sin 90^{\circ} \sin x

Since cos⁡90∘=0\cos 90^{\circ} = 0 and sin⁡90∘=1\sin 90^{\circ} = 1, we can simplify this expression to:

cos⁡(90∘+x)=−sin⁡x\cos \left(90^{\circ}+x\right) = -\sin x

Similarly, we can simplify the second term using the identity:

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A+B) = \sin A \cos B + \cos A \sin B

Applying this identity to the second term, we get:

sin⁡(180∘+x)=sin⁡180∘cos⁡x+cos⁡180∘sin⁡x\sin \left(180^{\circ}+x\right) = \sin 180^{\circ} \cos x + \cos 180^{\circ} \sin x

Since sin⁡180∘=0\sin 180^{\circ} = 0 and cos⁡180∘=−1\cos 180^{\circ} = -1, we can simplify this expression to:

sin⁡(180∘+x)=−sin⁡x\sin \left(180^{\circ}+x\right) = -\sin x

Now, we can multiply the two simplified terms to get:

cos⁡(90∘+x)⋅sin⁡(180∘+x)=(−sin⁡x)(−sin⁡x)\cos \left(90^{\circ}+x\right) \cdot \sin \left(180^{\circ}+x\right) = (-\sin x)(-\sin x)

=sin⁡2x= \sin^2 x

Simplifying the Denominator

The denominator of the expression is:

tan⁡225∘−cos⁡2(−x)\tan 225^{\circ} - \cos^2(-x)

We can simplify this expression by using the trigonometric identity:

tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A}

Applying this identity to the first term, we get:

tan⁡225∘=sin⁡225∘cos⁡225∘\tan 225^{\circ} = \frac{\sin 225^{\circ}}{\cos 225^{\circ}}

Since sin⁡225∘=−22\sin 225^{\circ} = -\frac{\sqrt{2}}{2} and cos⁡225∘=−22\cos 225^{\circ} = -\frac{\sqrt{2}}{2}, we can simplify this expression to:

tan⁡225∘=−22−22\tan 225^{\circ} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}

=1= 1

Now, we can simplify the second term using the identity:

cos⁡2A=1−sin⁡2A\cos^2 A = 1 - \sin^2 A

Applying this identity to the second term, we get:

cos⁡2(−x)=1−sin⁡2(−x)\cos^2(-x) = 1 - \sin^2(-x)

Since sin⁡(−x)=−sin⁡x\sin(-x) = -\sin x, we can simplify this expression to:

cos⁡2(−x)=1−sin⁡2x\cos^2(-x) = 1 - \sin^2 x

Now, we can substitute the simplified expressions into the denominator to get:

tan⁡225∘−cos⁡2(−x)=1−(1−sin⁡2x)\tan 225^{\circ} - \cos^2(-x) = 1 - (1 - \sin^2 x)

=sin⁡2x= \sin^2 x

Combining the Simplified Terms

Now that we have simplified the numerator and denominator, we can combine the terms to get:

cos⁡(90∘+x)⋅sin⁡(180∘+x)tan⁡225∘−cos⁡2(−x)=sin⁡2xsin⁡2x\frac{\cos \left(90^{\circ}+x\right) \cdot \sin \left(180^{\circ}+x\right)}{\tan 225^{\circ} - \cos^2(-x)} = \frac{\sin^2 x}{\sin^2 x}

=1= 1

Conclusion

In this article, we simplified a complex trigonometric expression without the use of a calculator. We broke down the expression into smaller components, applied trigonometric identities, and manipulated the terms to arrive at a simplified form. The final result is a simple expression that can be evaluated using basic trigonometric functions.

Final Answer

The final answer is 1\boxed{1}.

Introduction

In our previous article, we simplified a complex trigonometric expression without the use of a calculator. We broke down the expression into smaller components, applied trigonometric identities, and manipulated the terms to arrive at a simplified form. In this article, we will answer some of the most frequently asked questions related to the simplification of trigonometric expressions.

Q&A

Q: What is the most important thing to remember when simplifying trigonometric expressions?

A: The most important thing to remember when simplifying trigonometric expressions is to use trigonometric identities to break down the expression into smaller components. This will make it easier to manipulate the terms and arrive at a simplified form.

Q: How do I know which trigonometric identity to use when simplifying an expression?

A: When simplifying an expression, you should look for opportunities to use trigonometric identities that involve the same trigonometric function. For example, if you have an expression that involves sine and cosine, you can use the identity sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1 to simplify the expression.

Q: What is the difference between a trigonometric identity and a trigonometric formula?

A: A trigonometric identity is a mathematical statement that is true for all values of the trigonometric function, while a trigonometric formula is a specific expression that involves trigonometric functions. For example, the identity sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1 is true for all values of A, while the formula sin⁡A=opphyp\sin A = \frac{opp}{hyp} is a specific expression that involves the sine function.

Q: How do I know when to use the Pythagorean identity?

A: The Pythagorean identity sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1 is a useful tool for simplifying expressions that involve sine and cosine. You should use this identity when you have an expression that involves both sine and cosine, and you want to simplify it.

Q: What is the difference between a trigonometric expression and a trigonometric equation?

A: A trigonometric expression is a mathematical statement that involves trigonometric functions, while a trigonometric equation is a mathematical statement that involves trigonometric functions and is equal to zero. For example, the expression sin⁡A+cos⁡A\sin A + \cos A is a trigonometric expression, while the equation sin⁡A+cos⁡A=0\sin A + \cos A = 0 is a trigonometric equation.

Q: How do I know when to use the sum and difference formulas?

A: The sum and difference formulas are useful tools for simplifying expressions that involve sine and cosine. You should use these formulas when you have an expression that involves the sum or difference of two angles.

Q: What is the difference between a trigonometric function and a trigonometric ratio?

A: A trigonometric function is a mathematical function that involves trigonometric ratios, while a trigonometric ratio is a specific expression that involves the ratio of two sides of a right triangle. For example, the function sin⁡A\sin A is a trigonometric function, while the ratio opphyp\frac{opp}{hyp} is a trigonometric ratio.

Conclusion

In this article, we answered some of the most frequently asked questions related to the simplification of trigonometric expressions. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in simplifying trigonometric expressions.

Final Answer

The final answer is 1\boxed{1}.