\(\text { If } \tan ^{2} \theta=1-a^{2} \text { prove that }\sec \theta+\tan ^{3} \theta \operatorname{cosec} \theta=\left(2-a^{2}\right) \frac{3}{2}\)

\(\text { If } a \cos \theta+b \sin \theta=m \ \text { and } a \sin \theta-b \cos \theta=n \text {. }\text { prove that } a^{2}+b^{2}=m^{2}+n^{2}\)

\(\text { If } \operatorname{cosec} \theta-\sin \theta=m \text { and } \sec \theta-\cos \theta=\mathbf{n},\text { prove that }\left(m^{2} n\right)^{\frac{2}{3}}+\left(m n^{2}\right)^{\frac{2}{3}}=1 \text {. }\)

\(\text { If } \tan \mathrm{A}=\mathrm{n} \tan \mathrm{B} \text { and } \sin \mathrm{A}=\mathrm{m} \sin \mathrm{B} \text {, Prove }\text { that } \cos ^{2} \mathrm{~A}=\frac{m^{2}-1}{n^{2}-1} \text {. }\)

A tree is broken by the wind. the top struck the ground at an angle of 30^o and at a distance of 30 m from the root. Find the whole height of the tree.

The shadow of a vertical tower on level ground increases by 10 m, when the altitude of the sun changes from angle of elevation 45^o to 30^o. Find the height of the tower, correct to one place of decimal \((\sqrt{3}=1.732)\)

An observed from the top of a light house 100 m high above sea level, the angle of depression of a ship sailing directly towards it, changes from 30^o to 60^o. Determine the distances travelled by the ship during the period of observation \([\sqrt{3}=1.732]\)

From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30^o and 60^o respectively.
Find (i) The horizontal distance between the building and the lamp post. (ii) The height of the lamp post \((\sqrt{3}=1.732)\)

A pole 5 m high is fixed on the top of a tower The angle of elevation of the top of the pole observed from a point A on the ground is 60^o and the angle of depression of the point A from the top of the tower is 45^o. Find the height of the tower.

From a point 100 m above a lake the angle of elevation of a stationary helicopter is 30^o and the angle of depression of reflection of the helicopter in the lake is 60^o. Find the height of the helicopter.

A man standing on the deck of a ship, which is 10 m above water level. He observes the angle of elevation of the top of a hill as 60^o and the angle of depression of the base of the hill as 30^o. Calculate the distance of the hill from the ship and the height of the hill.