Explanation

Dividing cos²x + cos⁴x = 1 by cos²x (cos x > 0 here) gives 1 + cos²x = sec²x = 1 + tan²x, so cos²x = tan²x and hence sin²x = cos⁴x. Let u = sin x; then cos²x = (√5 − 1)/2 = c with c + c² = 1, and u² = c² gives u = c. The expression sin²x + 2sin³x + sin⁴x = (u + u²)² = (c + c²)² = 1, option D. A common error is to mis-expand the powers and conclude something else.