As
[tex]\tan x=\dfrac{\sin x}{\cos x}[/tex]
[tex]\cot x=\dfrac{\cos x}{\sin x}[/tex]
we have
[tex]\sin x(\tan x\cos x-\cot x\cos x)=\sin x\left(\sin x-\dfrac{\cos^2x}{\sin x}\right)=\sin^2x-\cos^2x[/tex]
Recalling that [tex]\cos2x=\cos^2x-\sin^2x[/tex], we end up with
[tex]-\cos2x=1-2\cos2x[/tex]
[tex]1=\cos2x[/tex]
[tex]\implies2x=2n\pi[/tex]
(where [tex]n[/tex] is any integer)
[tex]\implies x=n\pi[/tex]
