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 value of $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}$ is:

Option 1: $\sec\theta+\tan \theta$

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

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

Option 4: $\sec\theta-\tan \theta$

Question : The value of $\frac{3\left(\operatorname{cosec}^2 26^{\circ}-\tan ^2 64^{\circ}\right)+\left(\cot ^2 42^{\circ}-\sec ^2 48^{\circ}\right)}{\cot \left(22^{\circ}-\theta\right)-\operatorname{cosec}^2\left(62^{\circ}+\theta\right)-\tan \left(\theta+68^{\circ}\right)+\tan ^2\left(28^{\circ}-\theta\right)}$ is:

Option 1: 3

Option 2: 4

Option 3: –1

Option 4: –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 : If $\sec\theta-\tan\theta=\frac{1}{\sqrt3}$, then the value of $\sec\theta.\tan\theta$ is:

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

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

Option 3: $\frac{4}{\sqrt3}$

Option 4: $\frac{1}{\sqrt3}$