Trigonometry

goals
If $\displaystyle \sin { \left( A+B \right) } -\sin { A } .\cos { B } +\cos { A } .\sin { B }$, then the value of $\displaystyle \sin { { 75 }^{ o } }$ is :
A man standing on the bank of the river observes that the angle subtended by a tree on the opposite bank is $\displaystyle 60^{0}$. When he retires $36$ metres from the bank he finds the angle to be $\displaystyle 30^{0}$. The breadth of the river is
Find the general solution of: $2\cos^2 x+3\sin x=0$.
The expression $\dfrac { 1+\sin { 2\alpha } }{ \cos { \left( 2\alpha -2\pi \right) .\tan { \left( \alpha -\dfrac { 3\pi }{ 4 } \right) } } } -\dfrac { 1 }{ 4 } \sin { 2\alpha \left[ \cot { \dfrac { \alpha }{ 2 } +\cot { \left( \dfrac { 3\pi }{ 2 } +\dfrac { \alpha }{ 2 } \right) } } \right] }$, when simplified reduces to:
Solve that : $\dfrac { { 5cos }^{ 2 }60+{ 4sec }^{ 2 }30-{ tan }^{ 2 }45 }{ { sin }^{ 2 }30+{ cos }^{ 2 }30 }$