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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Question : If $0^{\circ}< \theta< 90^{\circ}$ and $\operatorname{cosec \theta} =\cot^{2}\theta$, then the value of expression $\operatorname{cosec^{4}\theta}–\operatorname{2cosec^{2}\theta}-\cot^{2}\theta$ is equal to:
Question : What is the value of $\frac{\cot \theta+\operatorname{cosec} \theta-1}{\cot \theta-\operatorname{cosec} \theta+1}$?
Question : If $\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{4}{5}$, then the value of $\frac{\operatorname{cosec}^2 \theta}{2-\operatorname{cosec}^2 \theta}$ is:
Question : If $\frac{\cos \theta}{(1+\sin \theta)}+\frac{\cos \theta}{(1-\sin \theta)}=4$ and $\theta$ is acute, then the value of $\theta$ is:
Question : Find the value of $\cos 0^{\circ}+\cos 30^{\circ}-\tan 45^{\circ}+\operatorname{cosec} 60^{\circ}+\cot 90^{\circ}$.
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