Question : A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height $h$ units. At a point on the plane, the angle of elevation at the bottom of the flagstaff is $\alpha$ and that of the top of the flagstaff is $\beta$. Then the height of the tower is:

Option 1: $h\tan\alpha$

Option 2: $\frac{h\tan\alpha}{(\tan\beta–\tan\alpha)}$

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

Option 4: None of these

Question : What is $\sin \alpha - \sin\beta$?

Option 1: $2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}$

Option 2: $2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}$

Option 3: $2 \cos \frac{\alpha-\beta}{2} \sin \frac{\alpha+\beta}{2}$

Option 4: $2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}$

Question : If $\frac{\cos \alpha}{\sin \beta} = 10$ and $\frac{\cos \alpha}{\cos \beta} = 11$, the value of $\cos ^2 \beta$ is:

Option 1: $\frac{121}{132}$

Option 2: $\frac{100}{221}$

Option 3: $\frac{88}{108}$

Option 4: $\frac{221}{121}$

Question : From two points, lying on the same horizontal line, the angles of elevation of the top of the pillar are $\theta$ and $\phi$ ($\theta<\phi$). If the height of the pillar is $h$ m and the two points lie on the same sides of the pillar, then the distances between the two points are:

Option 1: $h(\tan\theta-\tan\phi)$ metre

Option 2: $h(\cot\phi-\cot\theta)$ metre

Option 3: $h(\cot\theta-\cot\phi)$ metre

Option 4: $h\frac{(\tan\theta \tan\phi)}{(\tan\phi-\tan\theta)}$ metre

Question : If $\tan (\alpha+\beta)=a, \tan (\alpha-\beta)=b$, then the value of $\tan 2 \alpha$ is:

Option 1: $\frac{a+b}{1-a b}$

Option 2: $\frac{a+b}{1+a b}$

Option 3: $\frac{a-b}{1+a b}$

Option 4: $\frac{a-b}{1-a b}$