Question

prove that 1/tan3x+tanx-1/cot3x+cotx=cot4x

Answer 1

Answer

Answer 2

We will use the following identity :
\[(*)\[\tan(\alpha+\beta)=(\tan \alpha+\tan \beta)/(1-\tan \alpha \tan \beta).\]...\]
\[The L.H.S =1/(\tan 3x+\tan x)-1/(\cot 3x+\cot x),\]
\[=1/(\tan 3x+\tan x)-1/\left\{ 1/\tan 3x +1/\tan x \right\},\]
\[1/(\tan 3x+\tan x)-1/\left\{ (\tan 3x+\tan x)/(\tan 3x \tan x) \right\},\]
\[=1/(\tan 3x+\tan x) - (\tan 3x \tan x)/(\tan 3x+ \tan x),\]
\[=(1-\tan 3x \tan x)/(\tan 3x+\tan x),\]
\[=1/[(\tan 3x+\tan x)/(1-\tan 3x \tan x)],\]
\[=1/[\tan(3x+x)]...[ by (*)],\]
\[=1/\tan 4x,\]
\[=\cot 4x,\]
\[=The R.H.S.\]
\[Hence, the Proof.\]