Question

if x+y+z=180 then cos2x+cos2y-cos2z=

Answer 1

cos 2x + cos 2y = cos [(x+y) +( x-y )] + cos [ (x+y - (x-y)] 
        = 2 cos (x+y) cos (x-y)
        = 2 Cos (180 - z) cos (x-y) 
       =  - 2 cos z cos (x-y)

- 2 cos z cos (x-y) -  cos 2 z =  - 2 cos z cos (x-y) - 2 cos² z + 1
           = -2 cos z ( cos (x-y) + cos z ) + 1
         = -2 cos z [ cos (x-y) - cos (x+y) ] + 1
        = - 2 cos z [ 2 sin x sin y ] + 1
      = 1 - 4 sin x sin y cos z