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Chapter 5: Problem 7

Verify each identity. $$\sec x-\sec x \sin ^{2} x=\cos x$$

Short Answer

Expert verified
After simplifying the left side of the given equation using the Pythagorean identity, it becomes \(\cos x\). Thus, the identity \(\sec x - \sec x \sin^2 x = \cos x\) is verified as both sides of the equation are equal.

Step by step solution

01

Break down the left side

The given equation is \(\sec x - \sec x \sin^2 x = \cos x\). Note that \(\sec x = 1/ \cos x\). Thus, the left side can be rewritten as \(\frac{1}{\cos x} - \frac{1}{\cos x}\sin^2 x\)
02

Factor out \(\frac{1}{\cos x}\)

Factoring out \(\frac{1}{\cos x}\) from Step 1 will leave \(\frac{1}{\cos x}(1 - \sin^2 x)\)
03

Use Pythagorean Identity

The Pythagorean Identity states that \(\sin^2x + \cos^2x = 1\). Thus, \(1 - \sin^2x = \cos^2x\). Replacing \(1 - \sin^2x\) in the left side of our equation based on this identity, the left side of equation transforms to \(\frac{1}{\cos x}(\cos^2 x) = \cos x\)
04

Final Simplification

Solve \(\frac{1}{\cos x}(\cos^2 x)\) which equals to \(\cos x\). So, left side of equation equals to right side thus identity \(\sec x - \sec x \sin^2 x = \cos x\) is verified.

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