Question

sin x+sin 5x= cot 2x​

Answer 1

Step-by-step explanation:

1. sinx+sin(x+4x) =cot2x
2. 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