Simplify The Expression: ${ \frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{\cos \left(45^{\circ}-x\right) \cos X-\sin \left(45^{\circ}-x\right) \sin X} }$

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Introduction

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

Understanding the Expression

The given expression is:

sin⁑15βˆ˜β‹…cos⁑15∘cos⁑(45βˆ˜βˆ’x)cos⁑xβˆ’sin⁑(45βˆ˜βˆ’x)sin⁑x\frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{\cos \left(45^{\circ}-x\right) \cos x-\sin \left(45^{\circ}-x\right) \sin x}

This expression involves the product of sine and cosine functions in the numerator and a difference of cosine and sine functions in the denominator. Our goal is to simplify this expression using trigonometric identities.

Using the Cosine of a Difference Identity

The cosine of a difference identity states that:

cos⁑(Aβˆ’B)=cos⁑Acos⁑B+sin⁑Asin⁑B\cos (A - B) = \cos A \cos B + \sin A \sin B

We can rewrite the denominator of the given expression using this identity:

cos⁑(45βˆ˜βˆ’x)cos⁑xβˆ’sin⁑(45βˆ˜βˆ’x)sin⁑x=cos⁑(45βˆ˜βˆ’x)cos⁑xβˆ’sin⁑(45βˆ˜βˆ’x)sin⁑x\cos \left(45^{\circ}-x\right) \cos x-\sin \left(45^{\circ}-x\right) \sin x = \cos (45^{\circ}-x) \cos x - \sin (45^{\circ}-x) \sin x

=cos⁑45∘cos⁑x+sin⁑45∘sin⁑xβˆ’sin⁑45∘cos⁑xβˆ’cos⁑45∘sin⁑x= \cos 45^{\circ} \cos x + \sin 45^{\circ} \sin x - \sin 45^{\circ} \cos x - \cos 45^{\circ} \sin x

=cos⁑45∘(cos⁑xβˆ’sin⁑x)+sin⁑45∘(sin⁑xβˆ’cos⁑x)= \cos 45^{\circ} (\cos x - \sin x) + \sin 45^{\circ} (\sin x - \cos x)

Using the Sine and Cosine of a Sum Identity

The sine and cosine of a sum identity states that:

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

cos⁑(A+B)=cos⁑Acos⁑Bβˆ’sin⁑Asin⁑B\cos (A + B) = \cos A \cos B - \sin A \sin B

We can rewrite the expression using these identities:

cos⁑45∘(cos⁑xβˆ’sin⁑x)+sin⁑45∘(sin⁑xβˆ’cos⁑x)=cos⁑45∘cos⁑xβˆ’cos⁑45∘sin⁑x+sin⁑45∘sin⁑xβˆ’sin⁑45∘cos⁑x\cos 45^{\circ} (\cos x - \sin x) + \sin 45^{\circ} (\sin x - \cos x) = \cos 45^{\circ} \cos x - \cos 45^{\circ} \sin x + \sin 45^{\circ} \sin x - \sin 45^{\circ} \cos x

=cos⁑45∘cos⁑xβˆ’sin⁑45∘cos⁑x+sin⁑45∘sin⁑xβˆ’cos⁑45∘sin⁑x= \cos 45^{\circ} \cos x - \sin 45^{\circ} \cos x + \sin 45^{\circ} \sin x - \cos 45^{\circ} \sin x

=cos⁑45∘(cos⁑xβˆ’sin⁑x)βˆ’sin⁑45∘(cos⁑xβˆ’sin⁑x)= \cos 45^{\circ} (\cos x - \sin x) - \sin 45^{\circ} (\cos x - \sin x)

Simplifying the Expression

Now that we have rewritten the denominator using the cosine of a difference identity and the sine and cosine of a sum identity, we can simplify the expression:

sin⁑15βˆ˜β‹…cos⁑15∘cos⁑(45βˆ˜βˆ’x)cos⁑xβˆ’sin⁑(45βˆ˜βˆ’x)sin⁑x=sin⁑15βˆ˜β‹…cos⁑15∘cos⁑45∘(cos⁑xβˆ’sin⁑x)βˆ’sin⁑45∘(cos⁑xβˆ’sin⁑x)\frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{\cos \left(45^{\circ}-x\right) \cos x-\sin \left(45^{\circ}-x\right) \sin x} = \frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{\cos 45^{\circ} (\cos x - \sin x) - \sin 45^{\circ} (\cos x - \sin x)}

=sin⁑15βˆ˜β‹…cos⁑15∘(cos⁑45βˆ˜βˆ’sin⁑45∘)(cos⁑xβˆ’sin⁑x)= \frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{(\cos 45^{\circ} - \sin 45^{\circ}) (\cos x - \sin x)}

Using the Pythagorean Identity

The Pythagorean identity states that:

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

We can rewrite the expression using this identity:

sin⁑15βˆ˜β‹…cos⁑15∘(cos⁑45βˆ˜βˆ’sin⁑45∘)(cos⁑xβˆ’sin⁑x)=sin⁑15βˆ˜β‹…cos⁑15∘2β‹…2(cos⁑xβˆ’sin⁑x)\frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{(\cos 45^{\circ} - \sin 45^{\circ}) (\cos x - \sin x)} = \frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{\sqrt{2} \cdot \sqrt{2} (\cos x - \sin x)}

=sin⁑15βˆ˜β‹…cos⁑15∘2(cos⁑xβˆ’sin⁑x)= \frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{2 (\cos x - \sin x)}

Using the Sine and Cosine of a Difference Identity

The sine and cosine of a difference identity states that:

sin⁑(Aβˆ’B)=sin⁑Acos⁑Bβˆ’cos⁑Asin⁑B\sin (A - B) = \sin A \cos B - \cos A \sin B

cos⁑(Aβˆ’B)=cos⁑Acos⁑B+sin⁑Asin⁑B\cos (A - B) = \cos A \cos B + \sin A \sin B

We can rewrite the expression using these identities:

sin⁑15βˆ˜β‹…cos⁑15∘2(cos⁑xβˆ’sin⁑x)=sin⁑15βˆ˜β‹…cos⁑15∘2cos⁑(xβˆ’45∘)\frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{2 (\cos x - \sin x)} = \frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{2 \cos (x - 45^{\circ})}

Simplifying the Expression

Now that we have rewritten the expression using the sine and cosine of a difference identity, we can simplify it:

sin⁑15βˆ˜β‹…cos⁑15∘2cos⁑(xβˆ’45∘)=sin⁑15βˆ˜β‹…cos⁑15∘2cos⁑(45βˆ˜βˆ’x)\frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{2 \cos (x - 45^{\circ})} = \frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{2 \cos (45^{\circ} - x)}

Conclusion

In this article, we have simplified the given expression using trigonometric identities. We have used various identities, including the cosine of a difference identity, the sine and cosine of a sum identity, the Pythagorean identity, and the sine and cosine of a difference identity. By applying these identities, we have simplified the expression to its final form.

Final Answer

The final answer is:

sin⁑15βˆ˜β‹…cos⁑15∘2cos⁑(45βˆ˜βˆ’x)\frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{2 \cos (45^{\circ} - x)}

This is the simplified form of the given expression.

Introduction

In our previous article, we simplified the given expression using trigonometric identities. However, we received many questions from readers who wanted to know more about the process and the identities used. In this article, we will answer some of the most frequently asked questions about simplifying the expression.

Q: What is the cosine of a difference identity?

A: The cosine of a difference identity states that:

cos⁑(Aβˆ’B)=cos⁑Acos⁑B+sin⁑Asin⁑B\cos (A - B) = \cos A \cos B + \sin A \sin B

This identity is used to rewrite the expression in a simpler form.

Q: How do I apply the cosine of a difference identity?

A: To apply the cosine of a difference identity, you need to identify the terms in the expression that can be rewritten using this identity. In this case, we rewrote the denominator of the expression using this identity.

Q: What is the sine and cosine of a sum identity?

A: The sine and cosine of a sum identity states that:

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

cos⁑(A+B)=cos⁑Acos⁑Bβˆ’sin⁑Asin⁑B\cos (A + B) = \cos A \cos B - \sin A \sin B

These identities are used to rewrite the expression in a simpler form.

Q: How do I apply the sine and cosine of a sum identity?

A: To apply the sine and cosine of a sum identity, you need to identify the terms in the expression that can be rewritten using these identities. In this case, we rewrote the expression using these identities.

Q: What is the Pythagorean identity?

A: The Pythagorean identity states that:

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

This identity is used to rewrite the expression in a simpler form.

Q: How do I apply the Pythagorean identity?

A: To apply the Pythagorean identity, you need to identify the terms in the expression that can be rewritten using this identity. In this case, we rewrote the expression using this identity.

Q: What is the sine and cosine of a difference identity?

A: The sine and cosine of a difference identity states that:

sin⁑(Aβˆ’B)=sin⁑Acos⁑Bβˆ’cos⁑Asin⁑B\sin (A - B) = \sin A \cos B - \cos A \sin B

cos⁑(Aβˆ’B)=cos⁑Acos⁑B+sin⁑Asin⁑B\cos (A - B) = \cos A \cos B + \sin A \sin B

These identities are used to rewrite the expression in a simpler form.

Q: How do I apply the sine and cosine of a difference identity?

A: To apply the sine and cosine of a difference identity, you need to identify the terms in the expression that can be rewritten using these identities. In this case, we rewrote the expression using these identities.

Q: What is the final answer?

A: The final answer is:

sin⁑15βˆ˜β‹…cos⁑15∘2cos⁑(45βˆ˜βˆ’x)\frac{\sin 15^{\circ} \cdot \cos 15^{\circ}}{2 \cos (45^{\circ} - x)}

This is the simplified form of the given expression.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying the expression. We have explained the various identities used and provided examples of how to apply them. By following these steps, you can simplify complex expressions using trigonometric identities.

Final Tips

  • Always start by identifying the terms in the expression that can be rewritten using trigonometric identities.
  • Use the cosine of a difference identity to rewrite the expression in a simpler form.
  • Use the sine and cosine of a sum identity to rewrite the expression in a simpler form.
  • Use the Pythagorean identity to rewrite the expression in a simpler form.
  • Use the sine and cosine of a difference identity to rewrite the expression in a simpler form.

By following these tips, you can simplify complex expressions using trigonometric identities.

Additional Resources

  • Trigonometric identities: A comprehensive guide to trigonometric identities, including the cosine of a difference identity, the sine and cosine of a sum identity, the Pythagorean identity, and the sine and cosine of a difference identity.
  • Simplifying expressions: A step-by-step guide to simplifying expressions using trigonometric identities.
  • Trigonometric functions: A comprehensive guide to trigonometric functions, including sine, cosine, and tangent.

By following these resources, you can learn more about trigonometric identities and how to simplify complex expressions.