Proving the Trigonometric Identity: 2cos^2 x - 1 1 - 2sin^2 x cos^2 x - sin^2 x

In trigonometry, proving identities involves transforming one side of an equation into the other, demonstrating their equivalence. In this article, we will go through the steps to prove the identity 2cos^2 x - 1 1 - 2sin^2 x cos^2 x - sin^2 x.

Introduction

Trigonometric identities are fundamental in simplifying and solving complex trigonometric expressions. We will use the Pythagorean identity and the double angle formula to prove the given identity.

Step 1: Proving 2cos^2 x - 1 1 - 2sin^2 x

We start by working with the left-hand side (LHS) of the equation 2cos^2 x - 1 1 - 2sin^2 x.

Take the left-hand side:

LHS: 2cos^2 x - 1

Recall the Pythagorean identity:

sin^2 x cos^2 x 1

This implies:

sin^2 x 1 - cos^2 x

Substitute 1 - cos^2 x for sin^2 x in the right-hand side:

RHS: 1 - 2sin^2 x

Substitute sin^2 x 1 - cos^2 x:

1 - 2(1 - cos^2 x)

1 - 2 2cos^2 x

2cos^2 x - 1

This matches the LHS:

LHS: 2cos^2 x - 1 RHS: 2cos^2 x - 1

Step 2: Proving 1 - 2sin^2 x cos^2 x - sin^2 x

Now, we work with the equation 1 - 2sin^2 x cos^2 x - sin^2 x.

Take the left-hand side:

LHS: 1 - 2sin^2 x

Use the Pythagorean identity again:

cos^2 x 1 - sin^2 x

Substitute cos^2 x 1 - sin^2 x in the right-hand side:

RHS: cos^2 x - sin^2 x

Substitute cos^2 x 1 - sin^2 x:

(1 - sin^2 x) - sin^2 x

1 - 2sin^2 x

This matches the LHS:

LHS: 1 - 2sin^2 x RHS: 1 - 2sin^2 x

Conclusion

We have shown:

2cos^2 x - 1 1 - 2sin^2 x

1 - 2sin^2 x cos^2 x - sin^2 x

Thus, we conclude that:

2cos^2 x - 1 1 - 2sin^2 x cos^2 x - sin^2 x

This completes the proof.

Additional Insights

The double angle formula for cosine, cos 2x cos^2 x - sin^2 x, is a direct result of the above identity. The use of the Pythagorean identity in these proofs highlights the interconnectedness of trigonometric functions.

Practice Problems

1. Prove that cos^2 x - sin^2 x cos 2x using the result from the identity.

2. Simplify the expression 2cos^2 x - 1 and show that it equals cos 2x.

3. Explain the significance of using the Pythagorean identity in trigonometric proofs.

References

1. Weisstein, E. W. (2023). Pythagorean Identity. MathWorld.

2. Larson, R. (2021). Trigonometry (11th Edition). Cengage Learning.