Question

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

Answer 1

Simplification of
`(\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)=(\tan x-\tan y)/(1+\tan x \tan y)(\text { Equation } 1)`

By comparing the above equation 1 with
`(\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)=(\tan 9 x-\tan 5 x)/(1+\tan 9 x \tan 5 x)} \\\\ {\tan (4 x)=(\tan 9 x-\tan 5 x)/(1+\tan 9 x \tan 5 x)} \\\\ {\text { Therefore, } (\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,
`(\tan 9 x-\tan 5 x)/(1+\tan 9 x \tan 5 x)` is simplified as tan⁡4x