`(sin^(2)alpha-sin^(2)beta)/(sin alpha cos alpha-sin beta cos beta)=tan(alpha+beta)`

Prove that sin^(2)alpha + cos^(2) (alpha + beta) + 2 sin alpha sin beta cos (alpha + beta) is independent of alpha .

If x cos alpha + y sin alpha = x cos beta + y sin beta "then" (sin alpha - cos alpha - sin beta + cos beta)/(sin alpha + cos alpha - sin beta - cos beta)=

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

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

|(cos (alpha+beta),-sin (alpha+beta),cos 2 beta),(sin alpha, cos alpha, sin beta),(-cos alpha, sin alpha, cos beta)| is independent of

(cos alpha + cos beta)/( sin alpha - sin beta) + (sin alpha + sin beta)/( cos alpha - cos beta ) =

Find the value of f(alpha, beta) =|(cos (alpha + beta), -sin (alpha + beta), cos 2beta),(sin alpha, cos alpha, sin beta),(-cos alpha, sin alpha, cos beta)|