Question : What is $\tan \frac{\theta}{2}$?

Option 1: $\frac{\cos \theta}{1-\sin \theta}$

Option 2: $\frac{\sin \theta}{1-\cos \theta}$

Option 3: $\frac{\cos \theta}{1-\cos \theta}$

Option 4: $\frac{\sin \theta}{1+\cos \theta}$

Question : If $\cos \theta+\cos ^2 \theta=1$, find the value of $\sqrt{\sin ^4 \theta+\cos ^2 \theta}$.

Option 1: $\sqrt{2} \cos \theta$

Option 2: $2 \operatorname{cos} \theta$

Option 3: $\sqrt{2} \operatorname{sin} \theta$

Option 4: $2 \operatorname{sin} \theta$

Question : If $\cos\theta+\sin\theta=\sqrt{2}\cos\theta$, then $\cos\theta-\sin\theta$ is:

Option 1: $\sqrt{2}\tan\theta$

Option 2: $-\sqrt{2}\cos\theta$

Option 3: $-\sqrt{2}\sin\theta$

Option 4: $\sqrt{2}\sin\theta$

Question : Simplify the following expression.
$\frac{\sin \theta - 2 \sin ^3 \theta}{2 \cos ^3 \theta - \cos \theta}$

Option 1: $\tan \theta$

Option 2: $\sin \theta$

Option 3: $\sec \theta$

Option 4: $\cos \theta$

Question : The value of $\frac{\sin \theta+\cos \theta-1}{\sin \theta-\cos \theta+1} \times \sqrt{\frac{1+\sin \theta}{1-\sin \theta}}$ is:

Option 1: -2

Option 2: 2

Option 3: -1

Option 4: 1