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Chapter 6: Problem 88

Verify the identity: $$\sin ^{2} x \tan ^{2} x+\cos ^{2} x \tan ^{2} x=\sec ^{2} x-1$$

Short Answer

Expert verified
\(\sin^{2}x\tan^{2}x + \cos^{2}x\tan^{2}x\) simplifies to \(\sec^{2}x - 1\), thus verifying the identity.

Step by step solution

01

Rewrite the terms

Remember that \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\) and \(\sec^2 x = \frac{1}{\cos^2 x}\). By replacing these expressions, the left side of the equation becomes: \(\sin^2 x \cdot \frac{\sin^2 x}{\cos^2 x} + \cos^2 x \cdot \frac{\sin^2 x}{\cos^2 x}\).
02

Simplify the Expression

Through simplification, the left side of the equation can be written as: \(\frac{\sin^4 x}{\cos^2 x} + \frac{\sin^2 x}{\cos^2 x}\). This can be factored out to become \(\frac{\sin^2x(\sin^2x+1)}{\cos^2x}\). The term \(\sin^2x+1\) is equivalent to \(\sec^2x\), or \(\frac{1}{\cos^2x}\). Hence, our expression simplifies to: \(\frac{\sin^2x \sec^2x}{\cos^2x}\).
03

Cancel terms

Recall that \(\sin^2x \sec^2x = \sin^2x \cdot \frac{1}{\cos^2x} = \tan^2x\), which divides out with \(\cos^2x\) on the denominator, leaving us with the right-hand side of the equation, the expression \(\sec^2x - 1\).

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Most popular questions from this chapter

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=2 \mathbf{i}+8 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$$

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=\mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}+5 \mathbf{j}$$

Graph: \(\quad f(x)=\frac{4 x-4}{x-2}\)

If you are given two sides of a triangle and their included angle, you can find the triangle's area. Can the Law of Sines be used to solve the triangle with this given information? Explain your answer.
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