Question
Prove the following identities.
cot θ + tan θ = sec θ cosec θ
cot θ + tan θ = sec θ cosec θ
✓
Answer
L.H.S. $=\cot \theta + \tan \theta$
$= \cos \theta / \sin \theta + \sin \theta / \cos \theta$
$= (\cos^2 \theta + sin^2 \theta) / (\sin \theta \cos \theta)$ ..$\left[\cos ^2 \theta+\sin ^2 \theta=1\right]$
$= 1/(\sin \theta \cos \theta)$
$=\sec \theta \cdot \operatorname{cosec} \theta=\text { R.H.S. }$
$\therefore \cot \theta+\tan \theta=\sec \theta \operatorname{cosec} \theta$
$= \cos \theta / \sin \theta + \sin \theta / \cos \theta$
$= (\cos^2 \theta + sin^2 \theta) / (\sin \theta \cos \theta)$ ..$\left[\cos ^2 \theta+\sin ^2 \theta=1\right]$
$= 1/(\sin \theta \cos \theta)$
$=\sec \theta \cdot \operatorname{cosec} \theta=\text { R.H.S. }$
$\therefore \cot \theta+\tan \theta=\sec \theta \operatorname{cosec} \theta$
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