cos-(i)cos x + cosy-2cos2

1.	cos-(i)cos x + cosy-2cos2
Answer» By trigonometric sum identity, cos(A+B) = cos(A)cos(B) - sin(A)sin(B) andcos(A-B) = cos(A)cos(B) + sin(A)sin(B) ii) Adding both above, cos(A+B) + cos(A-B) = 2cos(A)cos(B) iii) Now, let A = (x + y)/2 and B = (x - y)/2 ==> A + B = (x + y + x - y)/2 = 2x/2 = x and A - B = (x + y - x + y)/2 = 2y/2 = y iv) Thus replacing these in (ii) above, cos(x) + cos(y) = 2cos[(x + y)/2]cos[(x - y)/2] [Proved]

Answer»

By trigonometric sum identity,

cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B) andcos(A-B) = cos(A)*cos(B) + sin(A)*sin(B)

ii) Adding both above, cos(A+B) + cos(A-B) = 2cos(A)*cos(B)

iii) Now, let A = (x + y)/2 and B = (x - y)/2

==> A + B = (x + y + x - y)/2 = 2x/2 = x

and A - B = (x + y - x + y)/2 = 2y/2 = y

iv) Thus replacing these in (ii) above,

cos(x) + cos(y) = 2cos[(x + y)/2]*cos[(x - y)/2] [Proved]

Discussion