In a triangle ABC, write \(\cos { \left( \frac { B+C }{ 2 } \right) } \) in terms of angle A.

If tan 2A = cot (A + 60), find the value of A where 2A is an acute angle.

If cos 2A = sin (A - 15), find A.

If \(\tan { \theta } =\frac { 1 }{ \sqrt { 3 } } ,\) evaluate \(\frac { { cosec }^{ 2 }\theta -\sec ^{ 2 }{ \theta } }{ { cosec }^{ 2 }\theta +\sec ^{ 2 }{ \theta } } .\)

Find the value of cot 10°. cot 30°. cot 80°

An observer, 1.7 m tall, is 20.J3 m away from a tower.The angle of elevation from the eye of observer to the top of tower is.30°. Find the height of tower

Find the length of kite string flying at 100 m above the ground with the elevation of 60⁰

If the angles of elevation of the top of a tower from two points distant a and b (a > b) from its foot and in the same straight line from it are respectively 30° and 60°, then find the height of the tower.

If the altitude of the Sun is 60°, what is the height of a tower which casts a shadow of length 30m?

Prove that : \(-1+\frac { \sin { A } \sin { \left( { 90 }^{ ° }-A \right) } }{ \cot { \left( { 90 }^{ ° }-A \right) } } =-\sin ^{ 2 }{ A } \)

Evaluate \(\frac { \tan { { 29 }^{ 0 } } }{ \cot { { 61 }^{ 0 } } } \)

In a \(\triangle\) ABC, right angles at B, if tan A=1, verify that 2sin A cos A=1.

If \(\cos { \theta } +\sin { \theta } =\sqrt { 2 } \cos { \theta } \), show that \(\cos { \theta } -\sin { \theta } =\sqrt { 2 } \sin { \theta } \).

From a point P on the ground, the angle of elevation of the top of a tower is 30° and that of the top of a flag staff fixed on the top of the tower is 60°. If the length of the flag staff is 5 m, find the height of the tower.

An aeroplane, when flying at a height of 4000m from the ground passes vertically above another aeroplane at an instant when the angles of elevation of the two planes from the same point on the ground are 60°and 45° respectively.Find the vertical distance between the aeroplanes at that instant. (Take √3 = 1.73)

A statue 1.6m tall stands on the top of a pedestal. From a point on the ground the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

If \(\sqrt { 3 } \sin { \theta } =\cos { \theta } \), find the value of \(\frac { \tan { \theta } (1+\cot { \theta } ) }{ \sin { \theta } +\cos { \theta } } .\)

In the given figure, \(\triangle ABC,\) is the right angles at B. \(\triangle BSC\) is right angles at S and BC = 7.5 cm, RS = 5 cm, RB = 6 cm \(\angle BSR=x^0\) and \(\angle SAB=y^0.\)  Find

(i) tan x⁰
(ii) sin y
(ii) cos y⁰

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
\({ (\sin { A } +cosecA) }^{ 2 }+{ (\cos { A } +\sec { A } ) }^{ 2 }=7+\tan ^{ 2 }{ A } +\cot ^{ 2 }{ A } \)

If (sec A + tan A)(sec B + tan B)(sec C + tan C) = (sec A - tan A)(sec B - tan B)(sec C - tan C), prove that each of the side is equal to \(\pm \) 1.