prove that cos^2 x + cos^2 y - 2cosx cosy cos (x+y) = sin^2(x+y) - Maths - Trigonometric Functions

Question

prove that cos^2 x + cos^2 y - 2cosx cosy cos (x+y) =
sin^2(x+y)

Vipin Verma · Accepted Answer

cos2 x + cos2 y - 2cosx cosy cos (x+y) =
sin2(x+y)

RHS = sin2 (x+y)
= [sin(x+y)]2
= [ sinx cosy + cosx siny ]2
= [ sin2xcos2y +
cos2xsin2y + 2sinx siny cosx cosy]
= (1-cos2x)cos2y +
(1-cos2y)cos2x + 2sinx siny cosx cosy
= cos2y -cos2xcos2y +
cos2x -cos2ycos2x + 2sinx siny
cosx cosy
= cos2 x + cos2 y -2cos2x
cos2y + 2sinx siny cosx cosy

And LHS = cos2 x + cos2 y - 2cosx cosy cos
(x+y)
= cos2 x + cos2 y - 2cosxcosy [cosx cosy
-sinx siny]
= cos2 x + cos2 y -2cos2x
cos2y + 2sinx siny cosx cosy
= RHS