Prove the Identity : \[ \cos x(\tan x+2)(2\tan x+1) = 2\sec x+5\sin x \]

Solution:

\[ \cos x(\tan x+2)(2\tan x+1) \]

\[ = \cos x(2\tan^2 x+5\tan x+2) \]

\[ = 2\cos x\tan^2 x + 5\cos x\tan x + 2\cos x \]

\[ = 2\cos x\cdot\frac{\sin^2 x}{\cos^2 x} + 5\sin x + 2\cos x \]

\[ = \frac{2\sin^2 x}{\cos x} + 5\sin x + 2\cos x \]

\[ = \frac{2(1-\cos^2 x)}{\cos x} + 5\sin x + 2\cos x \]

\[ = 2\sec x-2\cos x+5\sin x+2\cos x \]

\[ = 2\sec x+5\sin x \]

Hence proved.