Chapter 9.8, Problem 23E

To determine

The error in the given solution.

Answer to Problem 23E

  $\tan \left(x+\frac{\pi }{4}\right)=\frac{1+\tan x}{1-\tan x}$

Explanation of Solution

Given:

Solution,

  $\begin{array}{c}\tan \left(x+\frac{\pi }{4}\right)=\frac{\tan x+\tan \frac{\pi }{4}}{1+\tan x\tan \frac{\pi }{4}}\\ =\frac{\tan x+1}{1+\tan x}\\ =1\end{array}$

Formula used:

  $\tan \left(a+b\right)=\frac{\tan a+\tan b}{1-\tan a.\tan b}$

Calculation:

By the given expression,

  $\begin{array}{c}\tan \left(x+\frac{\pi }{4}\right)=\frac{\tan x+\tan \frac{\pi }{4}}{1-\tan x\tan \frac{\pi }{4}}\\ =\frac{\tan x+1}{1-\tan x}\\ =\frac{1+\tan x}{1-\tan x}\end{array}$

Conclusion:

Hence, the correct answerfor the given solution is $\frac{1+\tan x}{1-\tan x}$ .