Find the sum of the series `sin^2alpha+sin^2(alpha+beta)+sin^2(alpha+2beta)…+sin^2(alpha +(n-1) beta)`

Find the sum of the series sinalpha+sin(alpha+beta)+sin(alpha+2beta)+sin(alpha+3beta)+.....+sin(alpha+(n-1)beta)

Sum of n terms of the series sinalpha-sin(alpha+beta)+sin(alpha+2beta)-sin(alpha+3beta)+….

Sum of n terms of the series sinalpha-sin(alpha+beta)+sin(alpha+2beta)-sin(alpha+3beta)+….

Sum of n terms of the series sinalpha-sin(alpha+beta)+sin(alpha+2beta)-sin(alpha+3beta)+….

2sin^(2)beta+4cos(alpha+beta)sin alpha sin beta+cos2(alpha+beta)=

sin ^ (2) alpha + sin ^ (2) (alpha-beta) -2sin alpha cos beta sin (alpha-beta) = sin ^ (2)

Prove that: cos2 alpha cos2 beta+sin^(2)(alpha-beta)-sin^(2)(alpha+beta)=cos2(alpha+beta)

Prove that: cos2alpha\ cos2beta+sin^2(alpha-beta)-sin^2(alpha+beta)=cos2(alpha+beta) .

2 sin ^(2) beta + 4 cos (alpha + beta) sin alpha sin beta + cos 2 (alpha + beta )=