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Chapter 5: Problem 93

Determine whether the statement is true or false. Justify your answer. $$\tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x+1}{1-\tan x}$$

Short Answer

Expert verified
The statement is false. The correct expression for \(\tan \left(x-\frac{\pi}{4}\right)\) is \(\frac{\tan x - 1}{1+\tan x}\), not \(\frac{\tan x + 1}{1-\tan x}\).

Step by step solution

01

Recall Identity

Recall trigonometric identity for tangent of difference between two angles, \(\tan(x-y)=\frac{\tan x-\tan y}{1+\tan x\tan y}\). This will be used to simplify the problem formulation.
02

Apply Identity

Replacing \(y\) by \(\frac{\pi}{4}\) in the identity from step 1, simplifies to \(\tan(x-\frac{\pi}{4})=\frac{\tan x- \tan \frac{\pi}{4}}{1+\tan x\cdot \tan \frac{\pi}{4}}=\frac{\tan x - 1}{1+\tan x}\). Here we used the fact that \(\tan \frac{\pi}{4}=1\).
03

Compare Expressions

Comparing this result with the provided equation \(\frac{\tan x + 1}{1-\tan x}\), it is clear that these two are not the same. The signs of terms in both numerator and denominator are opposite in both expressions. Therefore, the statement is false.

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