Question : The expression $\frac{\left(1-2 \sin ^2 \theta \cos ^2 \theta\right)(\cot \theta+1) \cos \theta}{\left(\sin ^4 \theta+\cos ^4 \theta\right)(1+\tan \theta) \operatorname{cosec} \theta}-1,0^{\circ}<\theta<90^{\circ}$, equals:

Option 1: $\cos ^2 \theta$

Option 2: $-\sin ^2 \theta$

Option 3: $\sec ^2 \theta$

Option 4: $-\sec ^2 \theta$

Question : If $\left(\frac{\cos A}{1-\sin A}\right)+\left(\frac{\cos A}{1+\sin A}\right)=4$, then what will be the value of $A$? $\left(0^{\circ}<\theta<90^{\circ}\right)$

Option 1: $90^{\circ}$

Option 2: $45^{\circ}$

Option 3: $60^{\circ}$

Option 4: $30^{\circ}$

Question : $\cos \left(30^{\circ}+\theta\right)-\sin \left(60^{\circ}-\theta\right)=$ _____________.

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

Option 2: $0$

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

Option 4: $\frac{1}{\sqrt{2}}$

Question : $\left(\frac{\tan ^3 \theta}{\sec ^2 \theta}+\frac{\cot ^3 \theta}{\operatorname{cosec}^2 \theta}+2 \sin \theta \cos \theta\right) \div\left(1+\operatorname{cosec}^2 \theta+\tan ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, is equal to:

Option 1: $\operatorname{cosec} \theta \sec \theta$

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

Option 3: $\sin \theta \cos \theta$

Option 4: $\sec \theta$

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