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Chapter 5: Problem 16

Verify the identity. $$\sin ^{2} \alpha-\sin ^{4} \alpha=\cos ^{2} \alpha-\cos ^{4} \alpha$$

Short Answer

Expert verified
The given identity \( \sin^{2}\alpha - \sin^{4}\alpha = \cos^{2}\alpha - \cos^{4}\alpha \) is verified

Step by step solution

01

Convert the expression

We'll start with the left-hand side (LHS) of the identity, \( \sin^{2}\alpha - \sin^{4}\alpha \) Using the Pythagorean identity, \( \sin^{2}\alpha \) can be written as\( 1 - \cos^{2}\alpha \) . Substitute this in to get \( 1 - \cos^{2}\alpha - \sin^{4}\alpha \)
02

Simplify the expression

Now, replace \( \sin^{4}\alpha \) with \( (1 - \cos^{2}\alpha)^{2} \), \( 1 - \cos^{2}\alpha - (1 - \cos^{2}\alpha)^{2} \) This simplifies to \( \cos^{2}\alpha - \cos^{4}\alpha \)
03

Check the right-hand side

Since the LHS simplifies to \( \cos^{2}\alpha - \cos^{4}\alpha \), and the RHS of the identity is also \( \cos^{2}\alpha - \cos^{4}\alpha \), the identity is confirmed.

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