If \(cos\left( \alpha +\beta \right) =\frac { 4 }{ 5 } ,\quad sin(\alpha -\beta )=\frac { 5 }{ 13 } \) and \(\alpha , \beta \) lie between 0 and \(\frac { \pi }{ 4 } \) then prove that \(tan 2\alpha =\frac { 56 }{ 33 } .\)

4

(a)

If \(\theta +\phi =\alpha \) and \(tan \theta =k\) \(tan\quad \phi ,\) then prove that \(sin\left( \theta -\phi \right) =\frac { k-1 }{ k+1 } sin\alpha .\)

4

(a)

If tan(πcosθ)=cot(πsinθ)  then prove that  \(cos\left( \theta -\frac { \pi }{ 4 } \right) =\pm \frac { 1 }{ 2\sqrt { 2 } } .\)

4

(a)

If \(cos \left( \theta +\phi \right) = m\quad cos\quad \left( \theta -\phi \right) \)then find the value of \(\frac { 1-m }{ 1+m } cot\quad \phi .\)

4

(a)

Two trees A and B are on the same side of a river. From a point C in the river the distance of trees A and B are 250m and 300m, respectively of the angle C is 45⁰, find the distance between the trees.[use \(\sqrt { 2 } =1.44\)]

4

(a)