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Chapter 5: Problem 21

Verify the identity. $$\sin ^{1 / 2} x \cos x-\sin ^{5 / 2} x \cos x=\cos ^{3} x \sqrt{\sin x}$$

Short Answer

Expert verified
Upon simplifying both sides of the given equation, we find that both sides are equal, thereby verifying the given trigonometric identity

Step by step solution

01

Simplify the left side of the equation

Start by factoring out the common terms on the left side, which is \(\sin^{1/2}x \cos x\). The equation becomes: \[\sin^{1/2}x \cos x(1 - \sin^2x)\]
02

Use the Pythagorean identity

Use the Pythagorean identity to replace \(1-\sin^2x\) by \(\cos^2x\). The equation now changes to: \[\sin^{1/2}x \cos x \cdot \cos^2x\]
03

Simplify further the left side

The result obtained from step 2 can be simplified to: \[\sin^{1/2}x \cos^3 x\]
04

Simplify the right side of the equation

On the right side, rewrite \(\cos^3x \sqrt{\sin x}\) as: \[\sin^{1/2}x \cos^3 x\]
05

Compare both sides

The left side and the right side of the equation are now identical, which verifies the original identity.

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