Question : If $a(\tan\theta +\cot \theta)=1$, $\sin\theta +\cos\theta =b$ with $0^{\circ}< \theta < 90^{\circ}$, then a relation between $a$ and $b$ is:

Option 1: $b^{2}=2\left ( a+1 \right )$

Option 2: $b^{2}=2\left ( a-1 \right )$

Option 3: $2a=b^{2}-1$

Option 4: $2a=b^{2}+1$

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

Option 1: $ \sin \theta+\cos \theta$

Option 2: $\sin \theta \cos \theta$

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

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

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

Option 1: $\sin \theta$

Option 2: $2 \cos \theta$

Option 3: $\cot \theta$

Option 4: $2 \tan \theta$

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

Option 1: $\sin \theta$

Option 2: $\cos \theta$

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

Option 4: $\cot \theta$