Inverse Trigono...

If $$\sin^{-1} x+\sin^{-1} y=\dfrac{\pi}{3}$$ then the value of $$\cos^{-1}x+\cos^{-1}y$$ is equal to which of the following :

$$ \tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1}\left(\frac{x-y}{x+y}\right)=\dots $$

$$f ( x ) = \sin ^ { - 1 } \sqrt { \frac { \sqrt { 1 + x ^ { 2 } } - 1 } { 2 \sqrt { 1 + x ^ { 2 } } } } ,$$ then which of the following is (are) correct?

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

If for $$x < -1,\ cos^{-1}{\left(\dfrac{x^{2}-1}{x^{2}+1}\right)}+sin^{-1}{\left(\dfrac{2x}{1+x^{2}}\right)}-tan^{-1}{\left(\dfrac{2x}{x^{2}-1}\right)}=\dfrac{\pi}{3}$$, then $$x=$$

Solve $${ \tan }^{ -1 }\left(\cfrac { \cos x }{ 1+\sin x }\right) $$.

If $${ cos }^{ -1 }\sqrt { p } +{ cos }^{ -1 }\sqrt { 1-p } +{ cos }^{ -1 }\sqrt { 1-q } =\frac { 3\pi  }{ 4 } $$ then the value of q is equal to:

The value of $$\sin ^{-1}[\cot{(\sin ^{-1}\sqrt{(\frac{2 - \sqrt 3}{4})} + \cos ^{-1}\frac{\sqrt {12}}{4})} + \sec ^{-1}\sqrt 2]$$

The solution set of the equation $${ sin }^{ -1 }\sqrt { 1-{ x }^{ 2 } } +{ cos }^{ -1 }x={ cot }^{ -1 }\left( \frac { \sqrt { 1-{ x }^{ 2 } }  }{ x }  \right) -{ sin }^{ -1 }x$$ is 

Number of solutions of the equation
$$tan^{ -1 } \left( \dfrac { 1 }{ a-1 }\right) = tan^{ -1 } \left(\dfrac { 1 }{ x }\right) + tan^{ -1 } \left( \dfrac { 1 }{ a^{ 2 } x + 1}\right)$$