sin5(x)=sin(x)

Question

help!! im stuck its to find a general solution

Answer 1

sin⁵(x)=sin(x),

sin⁵(x)-sin(x)=0,

sin(x)(sin⁴(x)-1)=0,

sin(x)(sin²(x)-1)(sin²(x)+1)=0,

sin(x)(sin(x)-1)(sin(x)+1)(sin²(x)+1)=0.

So sin(x)=0 is a solution⇒x=nπ where n is an integer.

sin(x)=1 is a solution⇒x=2πn+π/2=½π(4n+1).

sin(x)=-1 is a solution⇒x=2πn+3π/2=½π(4n+3).

The last two solutions can be combined: x=π(2n+1±½).

SOLUTION x=nπ or π(2n+1±½).