If sin (y + z - x), sin (z + x - y), sin (x + y - z) are in A.P., then prove that tan x, tan y, tan z are also in A. P

If sin \(\alpha +\sin { \beta } =a\) and \(\cos { \alpha + } \cos { \beta } =b\) , then prove that \(\cos { \left( a-\beta \right) } =\frac { { a }^{ 2 }+{ b }^{ 2 }-2 }{ 2 } \)

If tan \(\alpha={{1}\over{7}},\sin\beta={{1}\over{\sqrt{10}}},\) Prove that \(\alpha+2\beta={{\pi}\over{4}}\) where \(0<\alpha<{{\pi}\over{2}}\) and \(0<\beta<{{\pi}\over{2.}}\)

If cosA = \(\frac{4}{5}\)and cosB = \(\frac{12}{13}\),\(\frac{3\pi}{3}<(A, B)<2 \pi,\) find the value of cos(A+B).

Solve \({\tan}^{-1}\left( \frac { x-1 }{ x-2 } \right) +{ \tan }^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)