Question : If $\frac{\sin \theta}{\cot \theta+\operatorname{cosec} \theta}=1$, then what is the value of $\theta$?
Option 1: $30^{\circ}$
Option 2: $90^{\circ}$
Option 3: $0^{\circ}$
Option 4: $45^{\circ}$
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Correct Answer: $90^{\circ}$
Solution :
$\frac{\sin \theta}{\cot \theta+\operatorname{cosec} \theta}=1$
⇒ $\frac{\sin \theta}{\frac{\cos \theta}{\sin \theta} +\frac{1}{\sin \theta}}=1$
⇒ $\frac{\sin \theta}{\frac{1+\cos \theta}{\sin \theta}}=1$
⇒ $\frac{\sin^2 \theta}{1+\cos \theta}=1$
⇒ $\sin^2 \theta = 1+\cos \theta$
⇒ $1-\sin^2 \theta+\cos \theta = 0$
⇒ $\cos^2 \theta + \cos \theta = 0$ ($\because$ $ \sin^2 \theta+\cos^2 \theta =1$ )
⇒ $\cos \theta(1+ \cos \theta) = 0$
⇒ $\cos \theta= 0$ or $\cos \theta = –1$
⇒ $\theta = 90^{\circ}$ or $180^{\circ}$
Hence, the correct answer is $90^{\circ}$.
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