Prove that `(cos alpha-cos beta)^2+(sin alpha-sin beta)^2=4sin^2((alpha-beta)/2)`

if xcosalpha+ysinalpha=k=xcosbeta+ysinbeta ,show that x/(cos((alpha+beta)/2))=y/(sin((alpha+beta)/2))=k/(cos((alpha-beta)/2))

Prove that cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4 cos((alpha+beta)/2)cos((beta+gamma)/2)cos((gamma+alpha)/2)

if tanalpha , tanbeta are the roots of x^2+px+q=0 ,show that sin^2(alpha+beta)+p sin(alpha+beta)cos(alpha+beta)+q cos^2(alpha+beta)=q

If alpha+beta+gamma=0 ,prove that cosalpha+cosbeta+cosgamma=4cos(alpha/2)cos(beta/2)cos(gamma/2)-1

Prove that cos^2(pi/4-theta)-sin^2(pi/4-theta)=sin 2theta

Prove that sin alpha+sin(alpha+2pi/3)+sin(alpha+4pi/3)=0

Prove that in case of expansion of solid beta = 2alpha .

Prove that |(alpha, beta, gamma),(alpha^(2), beta^(2), gamma^(2)),(beta+gamma , gamma + alpha, alpha + beta)| = (alpha-beta)(beta-gamma)(gamma-alpha)(alpha + beta + gamma) .

If secalpha+secbeta+secgamma=0 ,show that (cosalpha+cosbeta+cosgamma)^2=cos^2alpha+cos^2beta+cos^2gamma