Inverse Trigono...

If $$cos{  }^{ -1 }\frac { x }{ a } +cos{  }^{ -1 }\frac { y }{ b } =\alpha \quad then\quad \frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { 2xy }{ ab } cos\alpha +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =$$

What is the value of $$\sin^{-1} \left\{ {\cos(\sin^{-1} x)} \right\} +\cos^{-1} \left\{ {\sin (\cos^{-1} x)} \right\} $$ ?

The value of $$\displaystyle sin^{1}\left ( sin\dfrac{5\pi}{3} \right )$$ is ......

The value of $$\tan \left[ \sin ^ { - 1 } \left( \cos \left( \sin ^ { - 1 } x \right) \right) \right]$$ $$\tan \left[ \cos ^ { - 1 } \left( \sin \left( \cos ^ { - 1 } x \right) \right) \right]$$ , $$( x \in ( 0,1 ) )$$ is equal to

If $$x = \sin ^ { - 1 } ( \sin 10 ) \text { and } y = \cos ^ { - 1 } ( \cos 10 )$$ then $$y - x$$ is equal to: 

The sum of the solution of the equation $$2\sin^{-1}\sqrt {x^{2}+x+1}+\cos^{-1}\sqrt {x^{2}+x}\dfrac {3\pi}{2}$$ is

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

The product of all 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

If $$\tan^{-1}{\left(\dfrac{3a^{2}x^{3}}{a^{3}-3ax^{2}}\right)}=k\tan^{-1}{\left(\dfrac{x}{a}\right)}$$, then $$k=$$

If $$tan^{ -1 }\left( { sin }^{ 2 }\theta +2sin+2 \right) $$+$${ cot }^{ -1 }\left( { 4 }^{ { sec }^{ 2 } }+1 \right) =\frac { \pi  }{ 2 } $$ has solution for some $$\theta $$ and $$\phi $$ then