If \(\sin { \alpha =\frac { 1 }{ 2 } } \) then find  the value of \(3\sin { \alpha } -4\sin ^{ 3 }{ \alpha } \)

If \(\sec { \theta } .\sin { \theta } =0\), then find the value of \(\theta\)

A ladder, leaning against a wall, makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

The ratio of the length of a rod and its shadow is \(1:{1\over \sqrt3}\).What is the angle of elevation of the source of light?

The top of two poles of height 16m and 10 mare connected by a wire of length I metre.If wire makes an angle of 30° with the horizontal, then find l.

Find the angle of elevation of the top of the tower from the point on the ground which is 30 m away from the foot of the tower of height 10 √3 rn.

Eliminate  \(\theta\) from the following equation. \(x=a\sec { \theta } ,y=b\tan { \theta } \)

If \(\sqrt { 3 } \tan { \theta } =3\sin { \theta } ,\) then find the value of \(\sin ^{ 2 }{ \theta } -\cos ^{ 2 }{ \theta } .\)

If \(\sqrt { 2 } \sin { \theta } =1\), find the value of \(\sec ^{ 2 }{ \theta } -cosec^{ 2 }\theta \)

If \(\theta \) be an acute angle and 5cosec \(\theta \) = 7, then evaluate \(\sin { \theta } +\cos ^{ 2 }{ \theta } -1\)

Find the value of \(\frac { \cos { { 60 }^{ 0 } } +\sin { { 45 }^{ 0 } } -\cot { { 30 }^{ 0 } } }{ \tan { { 60 }^{ 0 } } +\sec { { 45 }^{ 0 } } -cosec{ 30 }^{ 0 } } .\)

Evaluate \(\frac { \sin { { 70 }^{ 0 } } }{ \cos { { 20 }^{ 0 } } } +\frac { cosec{ 36 }^{ 0 } }{ \sec { { 54 }^{ 0 } } } -\frac { 2\cos { { 43 }^{ 0 } } cosec{ 47 }^{ 0 } }{ \tan { { 10 }^{ 0 } } \tan { { 40 }^{ 0 } } \tan { { 50 }^{ 0 } } \tan { { 80 }^{ 0 } } } \)

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

Prove that : \(\left( cosec\theta -\sin { \theta } \right) \left( \sec { \theta } -\cos { \theta } \right) \left( \tan { \theta } +\cot { \theta } \right) =1\)

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 tan\(\theta =\frac {a}{b},\) find the value of sec \(\theta.\)

Evaluate \({ \left( \frac { \sin { { 25 }^{ 0 } } }{ \cos { { 65 }^{ 0 } } } \right) }^{ 2 }+{ \left( \frac { \tan { { 65 }^{ 0 } } }{ \cot { { 25 }^{ 0 } } } \right) }^{ 2 }-2\cos ^{ 2 }{ { 45 }^{ 0 } } .\)

If \(\sec { \theta } +\tan { \theta } =p\),  show that \(\sec { \theta } -\tan { \theta } =\frac { 1 }{ p } \) . Hence fin the value of \(\cos { \theta } \) and \(\sin { \theta } \) .

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
\({ (cosec\theta -\cot { \theta } ) }^{ 2 }=\frac { 1-\cos { \theta } }{ 1+\cos { \theta } } \)

If tan A  = a tan B and sin A = b sin B, prove that \(\cos ^{ 2 }{ A } =\frac { { b }^{ 2 }-1 }{ { a }^{ 2 }-1 } \)