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Chapter 5: Problem 63

Verify the identity. $$\cos ^{3} x \sin ^{2} x=\left(\sin ^{2} x-\sin ^{4} x\right) \cos x$$

Short Answer

Expert verified
After distributing and simplifying, the left hand side of the equation equals \(\sin^2x\cos{x} - \sin^4x\cos{x}\), which is the same as the right hand side of the original identity.

Step by step solution

01

Rewrite the Left Hand Side

Rewrite \(\cos ^{3} x \sin ^{2} x\) as \((\cos^2x\cos{x})\sin^2x\).
02

Apply the Pythagorean Identity

We know the Pythagorean identity is \(\sin^2x + \cos^2x = 1\). Solve for \(\cos^2x\) by subtracting \(\sin^2x\) from both sides to get \(\cos^2x = 1 - \sin^2x\). Substitute this into the expression from Step 1 to get \((1 - \sin^2x)\cos{x}\sin^2x\).
03

Simplify the Expression

Distribute \(\cos{x}\sin^2x\) across the brackets to get \(\sin^2x\cos{x} - \sin^4x\cos{x}\). This equals the right hand side of the original identity.

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