Prove by vector metod the following formula of plane trigonometry `cos(alpha-beta)=cosalpha cosbeta+sinalpha sinbeta`

Show by vector method that sin(alpha-beta)=sinalphacosbeta-cosalpha sinbeta.

(sinalpha cos beta+cos alpha sin beta)^2+(cos alpha cos beta-sin alpha sin beta)^2=1

Prove by vector methods that sin(alpha+beta)=sin alpha cos beta+cos alpha sin beta

If alpha+beta=90^@ and alpha:beta=2:1 then sinalpha:sinbeta=

If sin alpha=sinbeta and cos alpha=cosbeta, then

If sinalpha+sin beta=a and cos alpha+cos beta=b,then sin(alpha+beta)=

The expression (sinalpha+sinbeta)/(cosalpha+cosbeta) is equal to

By geometrical interpretation, prove that (i) sin(alpha+beta)=sin alpha cos beta+sinbeta cosalpha (ii) cos(alpha+beta)=cosalpha cosbeta -sin alpha sinbeta

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

If cosalpha+cosbeta=0=sinalpha+sinbeta , then cos2alpha+cos2beta=?