Trigonometric F...

Solve:

If $$\cos { \theta  } -\sin { \theta  } =\sqrt { 2 } \sin { \theta  } $$, then $$\cos { \theta  } +\sin { \theta  } $$ is

$$\tan { \theta  } \left( 1-\cot ^{ 2 }{ \theta  }  \right) $$ is equal to

The value of 
$$\cos { \left( \pi /5 \right)  } \cos { \left( 2\pi /5 \right)  } \cos { \left( 4\pi /5 \right)  } \cos { \left( 8\pi /5 \right)  } $$ is 

If $$2\cos A+3\cos  B+5\cos C$$$$=2\sin A+3\sin B+5\sin C=0$$ then
$$8\cos 3A+27\cos 3B+125\cos C$$$$=k\cos(A+B+C)$$  then $$k=$$

If $$\sin { \alpha  } =12/13\left( 0<\alpha <\pi /2 \right)$$ and
$$\cos { \beta  } =-\dfrac { 3 }{ 5 } \left( \pi <\beta <\dfrac { 3 }{ 2 } \pi  \right) $$, the value of $$\sin { \left( \alpha +\beta  \right)  }$$ is

If $$\sin \left(\sin^{-1}\dfrac{1}{5}+\cos ^{-1}x\right)=1$$, then find the value of $$x$$.

The maximum value of the expression $$\dfrac { 1 }{ \sin ^{ 2 }{ \theta  } +3\sin { \theta  } \cos { \theta  } +5\cos ^{ 2 }{ \theta  }  } $$ is 

The number of solutions of the pair of equations.
$$2\sin^2\theta -\cos 2\theta =0$$
$$2\cos^2\theta -3\sin \theta =0$$
in the interval $$[0, 2\pi]$$ is?

If $$\alpha, \beta, \gamma, \delta$$ are the smallest $$+$$ive angles in ascending order of magnitude which have their sines equal to a $$+$$ive quantity $$\lambda$$ then the value of $$4\sin \dfrac{\alpha}{2}+3\sin \dfrac{\beta}{2}+2\sin \dfrac{\gamma}{2}+\sin \dfrac{\delta}{2}=$$.