Answer to Question #143546 in Trigonometry for nonas

Question

Find at least one solution to the following equation: sin(x^2 −1)/ 1−sin(x^2 −1) = sin(x) + sin^2(x) + sin^3(x) + sin^4(x) +···

Answer 1

sin(x²-1)/(1-sin(x²-1))=sin(x)+sin²(x)+sin³(x)+sin⁴(x)+...

sin(x²-1)/(1-sin(x²-1))=sin(x)/(1-sin(x))

sin(x²-1)(1-sin(x))=sin(x)(1-sin(x²-1))

sin(x²-1)-sin(x²-1)sin(x)=sin(x)-sin(x²-1)sin(x)

sin(x²-1)-sin(x)--sin(x²-1)sin(x)+sin(x²-1)sin(x)=0

sin(x²-1)-sin(x)=0

2sin((x²-1-x)/2)cos((x²-1+x)/2)=0

sin((x²-1-x)/2)=0

(x²-1-x)/2=0

x²-x-1=0

x=(1±√5)/2