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Reduce (csc^2 x - sec^2 x) to an expression containing only tan x.

Question:

Reduce {eq}(csc^2 x - sec^2 x) {/eq} to an expression containing only {eq}tan x. {/eq}

Trigonometric Identities

To solve questions of the type given here, we need to make use of trigonometric identities. Here, we use two different identities which help us to first expand the expression and then simplify it.

Answer and Explanation:


We perform the required task as follows.

$$\begin{align} &\csc^2 x - \sec^2 x\\ =&(1+\cot^2x)-(\tan^2x+1)\\ =&\cot^2x-\tan^2x\\ =&\frac{1}{\tan^2x}-\tan^2x&&&&\left [ \because \cot x=\frac{1}{\tan x} \right ]\\ \end{align} $$


Learn more about this topic:

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Trigonometric Identities: Definition & Uses

from Honors Precalculus Textbook

Chapter 23 / Lesson 1
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