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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
The trigonometric identity is verified since \( (\sin^{1/2} x \cos x)(1 - \sin^2 x) \), through simplification, equals \( \cos ^{3} x \sqrt{\sin x} \) , matching the right-hand side.

Step by step solution

01

Recognize common factors

Observe that in the left-hand side, both terms have \( \sin^{1/2} x \) and \( \cos x \) as their common factors. So we can rewrite it as \( (\sin^{1/2} x \cos x)(1 - \sin^2 x) \) .
02

Use Pythagorean identity

From the Pythagorean identity, we know that \( \sin^2 x + \cos^2 x = 1 \). Therefore, we could replace \( (1 - \sin^2 x) \) with \( \cos^2 x \) since they are equivalent.
03

Simplify the expression

Substitute \( \cos^2 x \) into the expression from step 1. The left-hand side (LHS) becomes \( \sin^{1/2} x \cos x \cos^2 x \).
04

Apply power of power rule

When raising a power to a power in an exponent, multiply the exponents. Hence, \( \cos x \cos^2 x \) becomes \( \cos^3 x \) and the left-hand side (LHS) now becomes \( \sin^{1/2} x \cos^3 x \) which is similar to the right hand side (RHS) \( \cos^3 x \sqrt{\sin{x}} \).
05

Final verification

Recognize that \( \sin^{1/2} x \) in the LHS is equivalent to \( \sqrt{\sin x} \) in the RHS. Therefore, the LHS does match the RHS, which verifies the trigonometric identity

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