Question

(sin x)(tan x cos x - cot x cos x) = 1 - 2 cos2x

Answer 1

As


`\tan x=(\sin x)/(\cos x)`

`\cot x=(\cos x)/(\sin x)`

we have


`\sin x(\tan x\cos x-\cot x\cos x)=\sin x\left(\sin x-(\cos^2x)/(\sin x)\right)=\sin^2x-\cos^2x`

Recalling that
`\cos2x=\cos^2x-\sin^2x`, we end up with


`-\cos2x=1-2\cos2x`

`1=\cos2x`

`\implies2x=2n\pi`

(where
`n` is any integer)


`\implies x=n\pi`