Prove the following identities: ( +2)(2 +1)=2 +5 - Vidyadip

Question

Prove the following identities:
$\cos\text{x}(\tan\text{x}+2)(2\tan\text{x}+1)=2\sec\text{x}+5\sin\text{x}$

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Answer

$\text{L.H.S}=\cos\text{x}(\tan\text{x}+2)(2\tan\text{x}+1)$
$=\cos\text{x}\Big(\frac{\sin\text{x}}{\cos\text{x}}+2\Big)\Big(\frac{2\sin\text{x}}{\cos\text{x}}=1\Big)\Big(\because\tan\text{x}=\frac{\sin\text{x}}{\cos\text{x}}\Big)$
$=\cos\frac{(\sin\text{x}+2\cos\text{x})(2\sin\text{x}+\cos\text{x})}{\cos\text{x}.\cos\text{x}}$
$=\frac{(2\sin^2\text{x}+\sin\text{x}\cos\text{x}+4\sin\text{x}\cos\text{x}+2\cos^2\text{x})}{\cos\text{x}}$
$=\frac{2(\sin^2\text{x}+\cos^2\text{x})+5\sin\text{x}\cos\text{x}}{\cos\text{x}}$
$=\frac{2+5\sin\text{x}\cos\text{x}}{\cos\text{x}}$ $(\because\sin^2\text{x}+\cos^2\text{x})=1$
$=\frac{2}{\cos\text{x}}+\frac{5\sin\text{x}\cos\text{x}}{\cos\text{x}}$
$=2\sec\text{x}+5\sin\text{x}$
$=\text{R.H.S}$
$\text{Proved}$

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