Question : Simplify. $\frac{\sin8\theta \cos\theta - \sin6\theta \cos3\theta}{\cos2\theta \cos\theta - \sin3\theta \sin4\theta}$
Option 1: $\cot \theta$
Option 2: $\cot 2 \theta$
Option 3: $\tan \theta$
Option 4: $\tan 2 \theta$
Correct Answer: $\tan 2 \theta$
Solution : Given: $\frac{\sin8\theta \cos\theta - \sin6\theta \cos3\theta}{\cos2\theta \cos\theta - \sin3\theta \sin4\theta}$ = $\frac{2\sin8\theta \cos\theta - 2\sin6\theta \cos3\theta}{2\cos2\theta \cos\theta - 2\sin3\theta \sin4\theta}$ = $\frac{\sin(8\theta+\theta)+\sin(8\theta-\theta)-\sin(6\theta+3\theta)-\sin(6\theta-3\theta)}{\cos(2\theta+\theta)+\cos(2\theta-\theta)-\cos(3\theta-4\theta)+\cos(3\theta+4\theta)}$ = $\frac{\sin9\theta + \sin7\theta - \sin9\theta - \sin3\theta}{\cos3\theta + \cos\theta - \cos(-\theta)+cos7\theta}$ = $\frac{\sin7\theta-\sin3\theta}{\cos3\theta + \cos\theta - \cos\theta+cos7\theta}$ = $\frac{\sin7\theta-\sin3\theta}{\cos7\theta+cos3\theta}$ = $\frac{2\cos\frac{7\theta+3\theta}{2}\sin\frac{7\theta-3\theta}{2}}{2\cos\frac{7\theta+3\theta}{2}\cos\frac{7\theta-3\theta}{2}}$ = $\frac{\cos5\theta\sin2\theta}{\cos5\theta\cos2\theta}$ = $\frac{\sin2\theta}{\cos2\theta}$ = $\tan2\theta$ Hence, the correct answer is $\tan2\theta$.
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Question : Simplify $\frac{\cos ^4 \theta-\sin ^4 \theta}{\sin ^2 \theta}$.
Option 1: $1-\tan ^2 \theta$
Option 2: $\tan ^2 \theta-1$
Option 3: $\cot ^2 \theta-1$
Option 4: $1-\cot ^2 \theta$
Question : The value of $\left(2 \cos ^2 \theta-1\right)\left[\frac{1+\tan \theta}{1-\tan \theta}+\frac{1-\tan \theta}{1+\tan \theta}\right]$ is:
Option 1: $2$
Option 2: $0$
Option 3: $\frac{\sqrt{3}}{2}$
Option 4: $1$
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: $\tan \theta$
Question : Simplify the following: $\mathrm{(1+cot^2\theta)(1-cos\theta)(1+cos\theta)}$
Option 1: 1
Option 2: –5
Option 3: 3
Option 4: –3
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:
Option 1: $\frac{2}{\sqrt3}$
Option 2: $\frac{\sqrt3}{2}$
Option 3: $\frac{1}{2}$
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