Work int the left side. By reciprocal identities, \(\sec x=1/\cos x\) so:

\(\tan x\sec x\sin x=\tan x\times\frac{1}{\cos x}\times\sin x\)

\(\tan x\sec x\sin x=\tan x\times\frac{\sin x}{\cos x}\)

By quotient identity, \(\tan x=\frac{\sin x}{\cos x}\) so:

\(\tan x\sec x\sin x=\tan x\times\tan x\)

\(\tan x\sec x\sin x=\tan^{2}x\)

\(\tan x\sec x\sin x=\tan x\times\frac{1}{\cos x}\times\sin x\)

\(\tan x\sec x\sin x=\tan x\times\frac{\sin x}{\cos x}\)

By quotient identity, \(\tan x=\frac{\sin x}{\cos x}\) so:

\(\tan x\sec x\sin x=\tan x\times\tan x\)

\(\tan x\sec x\sin x=\tan^{2}x\)