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Chapter 5: Problem 22

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\sin ^{2} x \sec ^{2} x-\sin ^{2} x$$

Short Answer

Expert verified
The simplified form of the given trigonometric expression is \(\frac{\sin^4 x}{\cos^2 x}\).

Step by step solution

01

Identify Common Factors

We first identify that \(\sin^2x\) is a common factor in both terms of the expression, hence can factorize the given expression as: \(\sin ^{2} x (\sec ^{2} x - 1)\).
02

Simplify using Trigonometric Identities

Next, we notice that \(\sec^2 x - 1\) is the same as \(\tan^2x\), from trigonometric identities. Therefore, we can substitute this into the expression. Therefore, our expression becomes: \(\sin ^{2} x \cdot \tan ^{2} x\).
03

Substitute Identity

Now, we know that the \(\tan x = \frac{\sin x}{\cos x}\), substituting this in the expression gives: \(\sin ^{2} x \cdot \left(\frac{\sin x}{\cos x}\right)^{2}\), which simplifies to \(\frac{\sin^4 x}{\cos^2 x}\).

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Most popular questions from this chapter

Find the exact value of the expression. $$\frac{\tan (5 \pi / 6)-\tan (\pi / 6)}{1+\tan (5 \pi / 6) \tan (\pi / 6)}$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\sin \frac{x}{2}+\cos x=0$$

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Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\). $$\cos \left(x-\frac{\pi}{2}\right)-\sin ^{2} x=0$$
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