Question : The value of $\frac{2 \cos ^3 \theta-\cos \theta}{\sin \theta-2 \sin ^3 \theta}$ is:
Option 1: $\sec \theta$
Option 2: $\sin \theta$
Option 3: $\cot \theta$
Option 4: $\tan \theta$
Correct Answer: $\cot \theta$
Solution : $\frac{2 \cos ^3 \theta-\cos \theta}{\sin \theta-2 \sin ^3 \theta}$ $=\frac{\cos \theta (2 \cos^2 \theta - 1)}{\sin \theta (1 - 2 \sin^2 \theta)}$ We know that $\cos2 \theta = 1 - 2\sin^2 \theta= 2 \cos^2 \theta-1$, Substituting these identities into the expression, $=\frac{\cos \theta \cos2 \theta}{\sin \theta \cos2 \theta}$ $=\frac{\cos \theta}{\sin \theta} = \cot \theta$ Hence, the correct answer is $\cot \theta$.
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Question : Simplify $\frac{\cos ^4 \theta-\sin ^4 \theta}{\sin ^2 \theta}$.
Question : $\sqrt{\frac{1+\sin\theta}{1-\sin\theta}}+\sqrt{\frac{1-\sin\theta}{1+\sin\theta}}$ is equal to:
Question : Find the value of $\left(\tan ^2 \theta+\tan ^4 \theta\right)$.
Question : If $\frac{\cos \theta}{1-\sin \theta}+\frac{\cos \theta}{1+\sin \theta}=4,0^{\circ}<\theta<90^{\circ}$, then what is the value of $(\sec \theta+\operatorname{cosec} \theta+\cot \theta) ?$
Question : If $\frac{\tan\theta +\cot\theta }{\tan\theta -\cot\theta }=2, (0\leq \theta \leq 90^{0})$, then the value of $\sin\theta$ is:
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