Prove the identity (sin θ + cos θ) (tan θ + cot θ ) = sec θ + cos… - Vidyadip

Question

Prove the identity $(sin θ + cos θ) (tan θ + cot θ ) = sec θ + cosec θ$.

✓

Answer

L.H.S. = (sin θ + cos θ)(tan θ + cot θ)
$=(\sin \theta+\cos \theta)\left(\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right)$
$=(\sin \theta+\cos \theta)\left(\frac{\sin ^2 \theta+\cos ^2 \theta}{\cos \theta \sin \theta}\right)$
$=(\sin \theta+\cos \theta) \times \frac{1}{\sin \theta \cos \theta} [\because \sin^2\theta + \cos^2\theta = 1]$
$=\frac{\sin \theta+\cos \theta}{\cos \theta \sin \theta}$
$=\frac{\sin \theta}{\cos \theta \sin \theta}+\frac{\cos \theta}{\cos \theta \sin \theta}$
$=\frac{1}{\cos \theta}+\frac{1}{\sin \theta}$
$=\sec \theta+\operatorname{cosec} \theta$
= R.H.S
Hence proved.

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