Prove that cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1.

Prove that cos 4x = 1 - 8 sin²x cos²x.

Prove that cos 6x = 32 cos⁶x - 48 cos⁴x + 18 cos²x - 1.

If  \(sin\left( { sin }^{ -1 }\left( \frac { 1 }{ 5 } \right) +{ cos }^{ -1 }(x) \right) =1\)  then find the value of x

Prove that \({ cot }^{ -1 }\left[ \frac { \sqrt { 1+sinx } +\sqrt { 1-sinx } }{ \sqrt { 1+sinx } -\sqrt { 1-sinx } } \right] =\frac { x }{ 2 } \) where \(x\in \left( 0,\frac { \pi }{ 4 } \right) \)