Sin ѳ + sqrt(sin ѳ + sqrt(sin ѳ + sqrt(sin ѳ + ...........................to infinity = sec⁴α, then sin ѳ = ?

As per Question Sin ѳ + √(sin ѳ + √(sin ѳ + √(sin ѳ + ...........................to infinity = sec⁴α ..........(1) Let √(sin ѳ +√(sin ѳ + √(sin ѳ + ...........................to infinity = y ......(2) Squaring both the sides. sinѳ +√(sinѳ +√(sinѳ.....................infinity = y² Therfore , sinѳ + y = y² But y²= sec⁴α implies that y= sec²α {from equation (1) and (2) }Therfore sinѳ + sec²α= sec⁴α sinѳ =sec⁴α-sec²α sinѳ =sec²α(sec²α-1) sinѳ = sec²α*tan²α