Question

Simplify tan 9x - tan 5x / 1 + tan 9x tan 5x.

Answer 1

Simplification of $\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}$  is tan⁡ 4x

Solution:

By using sum-difference formula for tangent,  

Tangent of difference between two angles is written as,

$\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}(\text { Equation } 1)$

By comparing the above equation 1 with $\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}$   ,  

we get x = 9x and y = 5x

By substituting  x =  9x and y = 5x in equation 1,

$\begin{array}{l}{\tan (9 x-5 x)=\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}} \\\\ {\tan (4 x)=\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}} \\\\ {\text { Therefore, } \frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}=\tan 4 x}\end{array}$

Hence by using difference formula for tangent,  $\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}$  is simplified as tan⁡4x