Question

Prove the following identities (1-16)

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

Answer 1

$$\begin{gathered} \mathrm{LHS}=\cos x\left(\tan x+2\right)\left(2\tan x+1\right) \\ =\cos x\left(2{\tan }^{2}x+5\tan x+2\right) \\ =\cos x\left(\frac{2{\sin }^{2}x}{{\cos }^{2}x}+\frac{5\sin x}{\cos x}+2\right) \\ =\frac{2{\sin }^{2}x+5\sin x\cos x+2{\cos }^{2}x}{\cos x} \\ =\frac{2+5\sin x\cos x}{\cos x} \\ =2\sec x+5\sin x \\ =\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$