Prove the following identities (1-16)


Prove the following identities (1-16)

$\cos x(\tan x+2)(2 \tan x+1)=2 \sec x+5 \sin x$


$\mathrm{LHS}=\cos x(\tan x+2)(2 \tan x+1)$

$=\cos x\left(2 \tan ^{2} x+5 \tan x+2\right)$

$=\cos x\left(\frac{2 \sin ^{2} x}{\cos ^{2} x}+\frac{5 \sin x}{\cos x}+2\right)$

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

$=\frac{2+5 \sin x \cos x}{\cos x}$

$=2 \sec x+5 \sin x$

$=$ RHS

Hence proved.

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