Prove each of the following identities by transforming the l

Prove each of the following identities by transforming the left side to the right.

a) sin 2 =2 sin cos

b) cos 2t= cos2t-sin2t

Solution

sin2alpha= sin(alpha+alpha)

sin(A+B)=sinAcosB +cosAsinB

sin(alpha + alpha)= sin alpha cos alpha + cos alpha sin alpha=2sin alphacos alpha

cos 2alpha= cos(alpha + alpha)

cos(A+B)=cosAcosB-sinAsinB

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