Question

Prove the following

(i) cosx + cosy = 2cos((x + y)/2)cos((x - y)/2)

(ii) cosx - cosy = - 2sin((x + y)/2)sin((x - y)/2)

(iii) snx + siny = 2sin((x + y)/2)cos((x - y)/2)

(iv) sinx - siny = 2cos((x - y)/2)sin((x + y)/2)

Answer 1

We have 

cos(A + B) = cosAcosB-sinAsinB …(1) 

cos(A – B) = cos AcosB + sin Asin B … (2) 

Adding and subtracting (1) and (2) we get 

cos( A + B) + cos( A -B) = 2 cos A cos B … (3) 

cos( A + B) – cos (A -B) = – 2 sin A sin B … (4) 

Further sin(A + B) = sinAcosB + cosAsinB         …(5) 

sin(A-B) = sinAcosB-cosAsinB …(6) 

Adding and subtracting (5) and (6) we get 

sin( A + B) + sin( A – B) = 2 sin A cos B … (7) 

sin( A + B) – sin(A – B) = 2cosA sinB … (8) 

Let A + B = x and A-B = y

Substituting thes values of A and B in (3), (4), (7) and (8), we get

Keen eye; 

• 2 cos xcos y = cos(x + y) + cos(x – y) 
• -2 sin x sin y = cos(x + y) – cos(x – y) 
• 2sin x cos y = sin(x + y) 4 - sin(x - y) 
• 2cosx siny = sin(x + y)-sin(x - y)