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Question
Prove the following identities
Solution
\[\text{ LHS }= \cos x\left( \tan x + 2 \right)\left( 2\tan x + 1 \right)\]
\[ = \cos x\left( 2 \tan^2 x + 5\tan x + 2 \right)\]
\[ = \cos x\left( \frac{2 \sin^2 x}{\cos^2 x} + \frac{5\sin x}{\cos x} + 2 \right)\]
\[ = \frac{2 \sin^2 x + 5\sin x \cos x + 2 \cos^2 x}{\cos x}\]
\[ = \frac{2 + 5\sin x \cos x}{\cos x}\]
\[ = 2\sec x + 5\sin x \]
= RHS
Hence proved.
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