Prove that `|cosalpha-cosbeta|lt=|alpha-beta|`

If cosalpha+cosbeta=0=sinalpha+sinbeta , then prove that cos2alpha +cos2beta=-2cos(alpha +beta) .

Prove that |[cos alpha cos beta, cos alpha sin beta , sin alpha],[-sinbeta,cosbeta,0],[sinalpha cosbeta, sinalpha sinbeta, cos alpha]|=cos2alpha

Prove that: |(sinalpha, cosalpha, 1),(sinbeta, cosbeta, 1),(singamma, cosgamma, 1)|=sin(alpha-beta)+sin(beta-gamma)+sin(gamma-alpha)

If alpha+beta= 60^0 , prove that cos^2 alpha + cos^2 beta - cosalpha cos beta = 3/4 .

If cosalpha+cosbeta=0=sinalpha+sinbeta+siny, prove that (i) sumcos3alpha=3cos(alpha+beta+gamma) 2. sumsin3alpha=3sin(alpha+beta+gamma) 3. sumcos^2alpha=sumsin^2alpha=3/2

For all value of alpha,beta,gamma prove that ,cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos(alpha+beta)/2*cos(beta+gamma)/2*cos(gamma+alpha)/2

If the lines x(sinalpha+sinbeta)-ysin(alpha-beta)=3 and x(cosalpha+cosbeta)+ycos(alpha-beta)=5 are perpendicular then sin2alpha+sin2beta is equal to:

Prove that for all ninN Cosalpha+cos(alpha+beta)+cos(alpha+2beta)+ . . . +cos[alpha+(n-1)beta] = (cos[alpha+((n-1)/(2))beta]"sin"((nbeta)/(2)))/("sin"(beta)/(2))

If cosalpha=1/2(x+1/x)a n dcosbeta=1/2(y+1/y),(x y >0); x , y ,alpha,beta in R then sin(alpha+beta+gamma+singammaAAgamma in R cosalphacosbeta=1AAalpha,beta in R (cosalpha+cosbeta)^2=4AAalpha,beta in R sin(alpha+beta+gamma)=sinalpha+sinbeta+s ingammaAAa , b ,gamma in R

(1+cosalphacosbeta)^2-(cosalpha+cosbeta)^2=