Question

If sin A+ sin B= a and
cos A + cos B = b, find :
sin (A + B)
cos (A + B)
cos (A - B)​

Answer 1

 If sin A + sin B = a and cos A + cos B = b , find cos (A+B) in terms of a and b ?

a=sinA+sinBa=sin⁡A+sin⁡B

a2=sin2A+sin2B+2sinAsinBa2=sin2⁡A+sin2⁡B+2sin⁡Asin⁡B

b=cosA+cosBb=cos⁡A+cos⁡B

b2=cos2A+cos2B+2cosAcosBb2=cos2⁡A+cos2⁡B+2cos⁡Acos⁡B

b2+a2=2+2(cosAcosB+sinAsinB)=2+2cos(A−B)b2+a2=2+2(cos⁡Acos⁡B+sin⁡Asin⁡B)=2+2cos⁡(A−B)

cos(A−B)=b2+a2−22cos⁡(A−B)=b2+a2−22

b2−a2b2−a2

=cos2A−sin2A+cos2B−sin2B+2(cosAcosB−sinAsinB)=cos2⁡A−sin2⁡A+cos2⁡B−sin2⁡B+2(cos⁡Acos⁡B−sin⁡Asin⁡B)

=cos2A