Inverse Trigono...

the number o real soluttion of ${ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\frac { \pi  }{ 2 } is$

The numerical value of $cot\left( { 2sin }^{ -1 }\dfrac { 3 }{ 5 } +{ cos }^{ -1 }\dfrac { 3 }{ 5 }  \right)$ is

If ${ sin }^{ -1 }\left( \dfrac { \sqrt { x }  }{ 2 }  \right) +{ sin }^{ -1 }\left( \sqrt { 1-\dfrac { x }{ 4 }  }  \right) +tan^{ -1 }y=\dfrac { 2\pi  }{ 3 }$ then

${ cos }^{ -1 }\left[ 2cot^{ -1 }\left( \surd 2-1 \right)  \right]$ is equal to

If ${ \cot }^{ -1 }(\sqrt { \cos a } )-{ \tan }^{ -1 }(\sqrt { \cos a } )=x,$ then $\sin x$ is equal to

${ tan }^{ -1 }\left( \dfrac { \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1+{ -x }^{ 2 } }  }{ \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1{ -x }^{ 2 } }  }  \right) ,[x]\le \dfrac { 1 }{ \sqrt { 2 }  } ,is\quad equal\quad to$

If If $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = r ^ { 2 } ,$ then $tan^ { - 1 } \left( \frac { x y } { z r } \right) + \tan ^ { - 1 } \left( \frac { y z } { x r } \right) + \tan ^ { - 1 } \left( \frac { x z } { y r } \right)$ =

If $cos^{-1}x - cos^{-1}\dfrac{y}{2} = \alpha$ , then $4x^2 - 4xy\, cos\, \alpha + y^2$ is equal to -

If  $\cos ^ { - 1 } ( x / a ) + \cos ^ { - 1 } ( y / b ) = \alpha ,$  Then  $x ^ { 2 } / a ^ { 2 } + y ^ { 2 } / b ^ { 2 }$  is equal to:

If $sin^{-1}x+sin^{-1}y=\cfrac {2 \pi}3$, then cos^{-1}x+cos^{-1}y$$ equal to