Question : The expression $(\tan \theta+\cot \theta)(\sec \theta+\tan \theta)(1-\sin \theta), 0^{\circ}<\theta<90^{\circ}$, is equal to:

Option 1: $\sec \theta$

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

Option 3: $\cot \theta$

Option 4: $\sin \theta$

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: $\cot \theta$

Question : If $0\leq\theta\leq 90^{\circ}$ and $4\cos^{2}\theta-4\sqrt{3}\cos\theta+3=0$, then the value of $\theta$ is:

Option 1: $30^{\circ}$

Option 2: $45^{\circ}$

Option 3: $90^{\circ}$

Option 4: $60^{\circ}$

Question : The value of $\frac{3\left(\cot ^2 47^{\circ}-\sec ^2 43^{\circ}\right)-2\left(\tan ^2 23^{\circ}-\operatorname{cosec}^2 67^{\circ}\right)}{\operatorname{cosec}^2\left(68^{\circ}+\theta\right)-\tan \left(\theta+61^{\circ}\right)-\tan ^2\left(22^{\circ}-\theta\right)+\cot \left(29^{\circ}-\theta\right)}$ is:

Option 1: –1

Option 2: 1

Option 3: 5

Option 4: 0