Question : If $a \cot \theta+b \operatorname{cosec} \theta=p$ and $b \cot \theta+a \operatorname{cosec} \theta=q$ then $p^2-q^2$ is equal to.

Option 1: $a^2+b^2$

Option 2: $a^2-b^2$

Option 3: $b^2-a^2$

Option 4: $b-a$

Question : If $a \cot \theta+b \operatorname{cosec} \theta=p$ and $b \cot \theta+a \operatorname{cosec} \theta=q$, then $p^2-q^2$ is equal to ____.

Option 1: $b^2-a^2$

Option 2: $a^2-b^2$

Option 3: $b-a$

Option 4: $a^2+b^2$

Question : $\frac{(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)}{(\sec \theta+\tan \theta)(1-\sin \theta)}$ is equal to:

Option 1: $2 \sec \theta$

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

Option 3: $\operatorname{cosec} \theta$

Option 4: $\sec \theta$

Question : The expression $\frac{\cos ^4 \theta-\sin ^4 \theta+2 \sin ^2 \theta+3}{(\operatorname{cosec} \theta+\cot \theta+1)(\operatorname{cosec} \theta-\cot \theta+1)-2}, 0^{\circ}<\theta<90^{\circ}$, is equal to:

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

Option 2: $2 \sin \theta$

Option 3: $\sec \theta$

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

Question : If $\operatorname{cosec} \theta+\cot \theta=p$, then the value of $\frac{p^2-1}{p^2+1}$ is:

Option 1: $\cos \theta$

Option 2: $\sin \theta$

Option 3: $\cot \theta$

Option 4: $\operatorname{cosec} \theta$

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: $\sec \theta$