Inverse Trigono...

If $$\displaystyle \sin ^{ -1 }{ x } +\sin ^{ -1 }{ y } +\sin ^{ -1 }{ z } =\frac { 3\pi  }{ 2 } $$ and $$f\left( 1 \right) =1,f\left( p+q \right) =f\left( p \right) .f\left( q \right) \quad \forall p,q\in R$$ then $$\displaystyle { x }^{ f\left( 1 \right)  }+{ y }^{ f\left( 2 \right)  }+{ z }^{ f\left( 3 \right)  }-\frac { x+y+z }{ { x }^{ f\left( 1 \right)  }+{ y }^{ f\left( 2 \right)  }+{ z }^{ f\left( 3 \right)  } } =$$

Exhaustive set of values of parameter $$a$$ so that $$\sin ^{ -1 }{ x } -\tan ^{ -1 }{ x } =a\quad $$ has a solution is

If $$\displaystyle \sum _{ i=1 }^{ 2n }{ \sin ^{ -1 }{ { x }_{ i } }  } =n\pi $$, then $$\displaystyle \sum _{ i=1 }^{ 2n }{ { x }_{ i } } $$ is equal to

The set of values of parameter $$a$$ so that the equation $$\displaystyle (\sin^{-1}x)^{3}+(\cos^{-1}x)^{3}=a\pi^{3}$$ has a solution. 

The number of real solutions of $$(x, y)$$, where is $$\displaystyle \:\left | y \right |= \sin x,y= \cos ^{-1}\left ( \cos x  \right),-2\pi \leq x\leq 2\pi ,$$ is

$$\displaystyle \:\cos ^{-1}\left \{ \dfrac{1}{2}x^{2}+\sqrt{1-x^{2}.}\sqrt{1-\dfrac{x^{2}}{4}} \right \}= \cos ^{-1}\dfrac{x}{2}-\cos ^{-1}x$$ holds for 

If $$\displaystyle \sin ^{ -1 }{ x } +\sin ^{ -1 }{ y } +\sin ^{ -1 }{ z } =\frac { 3\pi  }{ 2 } $$, then $$\displaystyle \frac { \sum _{ k=1 }^{ 2 }{ \left( { x }^{ 100k }+{ y }^{ 106k } \right)  }  }{ \sum { { x }^{ 207 }.{ y }^{ 207 } }  } $$ is

The product of all real values of x satisfying the equation
$$\sin^{-1}\cos \left ( \dfrac{2x^{2}+10\left | x \right |+4}{x^{2}+5\left | x \right |+3} \right )=\cot \left ( \cot ^{-1}\left ( \dfrac{2-18\left | x \right |}{9\left | x \right |} \right ) \right )+\dfrac{\pi }{2}$$ is

The value of $$a$$ for which $$\displaystyle ax^{2}+sin^{-1}(x^{2}-2x+2)+cos^{-1}(x^{2}-2x+2)=0$$ has a real solution is 

If $$\displaystyle sin^{-1}x+sin^{-1}y+sin^{-1}z=\pi$$, then $$x^{4}+y^{4}+z^{4}+4x^{2}y^{2}z^{2}=k(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2})$$, where $$k$$ is equal to