1- + 1- is equal to : - Vidyadip

MCQ

$\frac{\sin\theta}{1-\cot\theta}+\frac{\cos\theta}{1-\tan\theta}$ is equal to :

A
$0$
B
$1$
✓
$\sin\theta+\cos\theta$
D
$\sin\theta-\cos\theta$

✓

Answer

Correct option: C.

$\sin\theta+\cos\theta$

$\frac{\sin\theta}{1-\cot\theta}+\frac{\cos\theta}{1-\tan\theta}=\frac{\sin\theta}{1-\frac{\cos\theta}{\sin\theta}}+\frac{\cos\theta}{1-\frac{\sin\theta}{\cos\theta}}$
$=\frac{\sin\theta\times\sin\theta}{\sin\theta-\cos\theta}+\frac{\cos\theta\times\cos\theta}{\cos\theta-\sin\theta}$
$=\frac{\sin^2\theta}{\sin\theta-\cos\theta}-\frac{\cos^2\theta}{\sin\theta-\cos\theta}$
$=\frac{\sin^2\theta-\cos^2\theta}{\sin\theta-\cos\theta}$
$=\frac{(\sin\theta+\cos\theta)(\sin\theta-\cos\theta)}{\sin\theta-\cos\theta}$
$=\sin\theta+\cos\theta$

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