Question
Prove that:
cot θ + tan θ = cosec θ sec θ
cot θ + tan θ = cosec θ sec θ
✓
Answer
Taking LHS, and putting $\cot \theta=\frac{\cos \theta}{\sin \theta} \text { and } \tan \theta=\frac{\sin \theta}{\cos \theta}$
$=\cot \theta+\tan \theta$
$=\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}$
$=\frac{\cos ^2 \theta+\sin ^2 \theta}{\sin \theta \cos \theta}\left[A S, \sin ^2 \theta+\cos ^2 \theta=1\right]$
$=\frac{1}{\sin \theta \cos \theta}$
$=\operatorname{cosec} \theta \sec \theta$
= RHS
Proved !
$=\cot \theta+\tan \theta$
$=\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}$
$=\frac{\cos ^2 \theta+\sin ^2 \theta}{\sin \theta \cos \theta}\left[A S, \sin ^2 \theta+\cos ^2 \theta=1\right]$
$=\frac{1}{\sin \theta \cos \theta}$
$=\operatorname{cosec} \theta \sec \theta$
= RHS
Proved !
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