Sin a sin(a+B) + cos a cos(a+B); cos (a+B) + cos (a-B);

Question

Sin a sin(a+B) + cos a cos(a+B); cos (a+B) + cos (a-B);

Answer 1

sinAsin(A+B) + cosAcos(A+B)=cos(A-A-B)=cos(-B)=cosB

cos(A+B) + cos(A-B)=cosAcosB - sinAsinB + cosAcosB + sinAsinB = 2cosAcosB                  

Answer 2

sin a sin(a+B) + cos a cos(a+B); cos (a+B) + cos (a-B)=

=sin a (sin a cos b+cos a sin b)+ cos a (cos a cos b- sin a sin b)=

=sin^2a cos b+cos a sin b sin a+cos^2 a cos b -sin a sin b cos a=

=sin^2a cos b+ cos^2 a cos b=cos b(sin^2a+cos^2a)=cos b

cos (a+B) + cos (a-B)=

=cos a cos b - sin a sin b + cos a cos b + sin a sin b=

=cos a cos b+cos a cos b=2cos a cos b