Trigonometry

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 : $$\dfrac { { 5cos }^{ 2 }60+{ 4sec }^{ 2 }30-{ tan }^{ 2 }45 }{ { sin }^{ 2 }30+{ cos }^{ 2 }30 } $$