Sin x+sin 5x = cot 2x

Step-by-step explanation:

sinx+sin (x+4x) = cot2x sinx+sinx cos4x+cosx sin4x=cot2x

sinx+sinx cos4x=sinx (1+cos4x) =

sinx (1-1+2 (cos2x) ^2) = 2sinx (cos2x) ^2

sin4x=2sin2xcos2x

2sinx (cos2x) ^2+cosx+2sin2xcos2x=cot2x

2sinxcos2x (cos2x+2 (cosx) ^2) = cot2x

but 2 (cosx) ^2=cos2x+1

2sinxcos2x (cos2x+cos2x+1) = cos2x/sin2x

(2sinxsin2x) (2cos2x+1) = 1

(4cosx (sinx) ^2) (2cos2x+1) = 1