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Chapter 5: Problem 47

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\tan x-\frac{\sec ^{2} x}{\tan x}$$

Short Answer

Expert verified
The simplified form of the given expression is \(-1\).

Step by step solution

01

Substitute the Pythagorean Identity

Instead of \(\sec^2x\), substitute \(1 + \tan^2x\) in the equation. Thus, the equation becomes \(\tan x - \frac{1 + \tan^2x}{\tan x}\)
02

Simplify the Term

The expression \(\frac{1 + \tan^2x}{\tan x}\) can be split into two parts by dividing each term in the numerator by \(\tan x\). Therefore, \(\tan x - \frac{1 + \tan^2x}{\tan x}\) simplifies to \(\tan x - 1 - \tan x\)
03

Simplify Further

Finally, simplify the expression by subtracting \(\tan x - \tan x\) to get the short answer.

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