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: $\tan 2 \theta$

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 : If $\cos ^2 \theta-\sin ^2 \theta=\tan ^2 \phi$, then which of the following is true?

Option 1: $\cos \theta \cos \phi=1$

Option 2: $\cos ^2 \phi-\sin ^2 \phi=\tan ^2 \theta$

Option 3: $\cos ^2 \phi-\sin ^2 \phi=\cot ^2 \theta$

Option 4: $\cos \theta \cos \phi=\sqrt{2}$

Question : $\sqrt{\frac{1+\sin\theta}{1-\sin\theta}}+\sqrt{\frac{1-\sin\theta}{1+\sin\theta}}$ is equal to:

Option 1: $2\cos\theta$

Option 2: $2\sin\theta$

Option 3: $2\cot\theta$

Option 4: $2\sec\theta$

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:

Option 1: $\frac{2}{\sqrt3}$

Option 2: $\frac{\sqrt3}{2}$

Option 3: $\frac{1}{2}$

Option 4: $1$