Find the domain of f(x) = sin^-1 \((\frac{|x|-2}{3})+ \) cos^-1 \((\frac{1-|x|}{4})\)

Find the value of tan⁻¹(−1 ) + cos^-1\((\frac{1}{2})+sin^-1(-\frac{1}{2})\)

If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)

Prove that tan^-1 x + tan^-1 z = tan^-1\(\left[ \frac { x+y+z-xyz }{ 1-xy-yz-zx } \right] \)

If tan^-1 x + tan^-1y + tan^-1 z = \(\pi\), show that x + y + z = xyz

Prove that \({ tan }^{ -1 }x+{ tan }^{ -1 }\frac { 2x }{ 1-{ x }^{ 2 } } ={ tan }^{ -1 }\frac { 3x-{ x }^{ 3 } }{ 1-{ 3x }^{ 2 } } ,|x|<\frac { 1 }{ \sqrt { 3 } } \)

Simplify: \({ tan }^{ -1 }\frac { x }{ y } -{ tan }^{ -1 }\frac { x-y }{ x+y } \)

Find the number of solution of the equation tan^-1(x-1) +  tan^-1x + tan^-1(x + 1) = tan^-1(3x)

Find the value of \({ sin }^{ -1 }(-1)+{ cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) +{ cot }^{ -1 }(2)\)

Find the value of \({ cot }^{ -1 }(1)+{ sin }^{ -1 }\left( -\frac { \sqrt { 3 } }{ 2 } \right) -{ sec }^{ -1 }(-\sqrt { 2 } )\)

Find the value of  \(tan\left[ \frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 2a }{ 1+{ a }^{ 2 } } \right) +\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1-{ a }^{ 2 } }{ 1+{ a }^{ 2 } } \right) \right] \)