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Chapter 5: Problem 61

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 expression is \(\sin^{2}x( \frac{1}{1 - \sin^{2}x} - 1)\)

Step by step solution

01

Use the Reciprocal Identity

In the given expression \(\sin^{2}x\sec^{2}x-\sin^{2}x\), replace \(\sec^{2}x\) with \((\frac{1}{\cos^{2}x})\). This yields \(\sin^{2}x \frac{1}{\cos^{2}x}-\sin^{2}x\).
02

Rewrite Using Single Trigonometric Function

The expression can be simplified further to \(\frac{\sin^{2}x}{\cos^{2}x}-\sin^{2}x\). Notice that \(\frac{\sin^{2}x}{\cos^{2}x}\) is actually \(\tan^{2}x\). So the expression now becomes \(\tan^{2}x-\sin^{2}x\).
03

Rewrite \(\tan^{2}x\)

Now, we need to use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Since \(\tan^{2}x = \frac{\sin^{2}x}{\cos^{2}x}\), we can substitute \(\sin^{2}x + \cos^{2}x = 1\) into \(\cos^{2}x\), so that it becomes \(\frac{\sin^{2}x}{1 - \sin^{2}x}\). The expression then becomes \(\frac{\sin^{2}x}{1 - \sin^{2}x} - \sin^{2}x\).
04

Factoring Out Common Terms

Now factor out the common term \(\sin^{2}x\) from the expression. The final simplified expression will be \(\sin^{2}x( \frac{1}{1 - \sin^{2}x} - 1)\).

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