Proof of Trigonometric Identity: cos2B - sin2B tan2A Given cos2A - sin2A tan2B

Introduction

In trigonometry, proving identities is a fundamental aspect of understanding and manipulating trigonometric functions. In this article, we will go through a detailed step-by-step proof to show that cos^2 B - sin^2 B tan^2 A given the identity cos^2 A - sin^2 A tan^2 B. This proof involves utilizing various trigonometric identities and algebraic manipulations, making it a comprehensive exercise in trigonometry.

Step-by-Step Proof

1. **Pythagorean Identity**

The Pythagorean identity states that for any angle x,

sin^2 x cos^2 x 1.

From this, we can express sin^2 A and sin^2 B in terms of cosine as follows:

sin^2 A 1 - cos^2 A

sin^2 B 1 - cos^2 B

2. **Rearrange the Given Equation**

The given equation is:

cos^2 A - sin^2 A tan^2 B

Substituting for sin^2 A from the Pythagorean identity, we get:

cos^2 A - (1 - cos^2 A) tan^2 B

This simplifies to:

2cos^2 A - 1 tan^2 B

Hence, we can express tan^2 B in terms of cos^2 A:

tan^2 B 2cos^2 A - 1

3. **Express tan^2 A**

Using the identity tan^2 A frac{sin^2 A}{cos^2 A}, we substitute for sin^2 A:

tan^2 A frac{1 - cos^2 A}{cos^2 A} frac{1}{cos^2 A} - 1

4. **Using the Identity for cos2B - sin2B**

We know that:

cos^2 B - sin^2 B 2cos^2 B - 1

To relate this to tan^2 A using tan^2 B, we express cos^2 B from the identity:

tan^2 B frac{sin^2 B}{cos^2 B}

Thus,

cos^2 B frac{1}{1 tan^2 B}

5. **Substitute tan^2 B 2cos^2 A - 1**

Substituting the expression for tan^2 B in the equation for cos^2 B, we get:

cos^2 B frac{1}{1 2cos^2 A - 1} frac{1}{2cos^2 A}

6. **Calculating cos2B - sin2B**

Now, using sin^2 B 1 - cos^2 B:

sin^2 B 1 - frac{1}{2cos^2 A} frac{2cos^2 A - 1}{2cos^2 A}

Thus,

cos^2 B - sin^2 B frac{1}{2cos^2 A} - frac{2cos^2 A - 1}{2cos^2 A}

Simplifying,

cos^2 B - sin^2 B frac{1 - (2cos^2 A - 1)}{2cos^2 A} frac{2 - 2cos^2 A}{2cos^2 A} frac{1 - cos^2 A}{cos^2 A} tan^2 A

Conclusion

Therefore, we have shown that:

cos^2 B - sin^2 B tan^2 A

This completes the proof and demonstrates the relationship between the given trigonometric identities.