Question : The angle of elevation of the top of a tower, vertically erected in the middle of a paddy field, from two points on a horizontal line through the foot of the tower are given to be $\alpha$ and $\beta(\alpha>\beta)$. The height of the tower is $h$ units. A possible distance (in the same unit) between the points is:

Option 1: $\frac{h(\cot\beta–\cot\alpha)}{\cos(\alpha+\beta)}$

Option 2: $h(\cot\alpha–\cot\beta)$

Option 3: $\frac{h(\tan\beta–\tan\alpha)}{\tan\alpha\tan\beta}$

Option 4: $h(\cot\alpha+\cot\beta)$

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 : $(\sin \theta+\operatorname{cosec} \theta)^2+(\cos \theta+\sec \theta)^2=$?

Option 1: $5+\tan ^2 \theta+\cot ^2 \theta$

Option 2: $7+\tan ^2 \theta-\cot ^2 \theta$

Option 3: $7+\tan ^2 \theta+\cot ^2 \theta$

Option 4: $5+\tan ^2 \theta-\cot ^2 \theta$

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: $\cot \theta$