Sqrt (0,5+sin^2x) + cos2x=1

√ (0,5+sin²x) + cos2x=1

√ (0,5+sin²x) = sin²x+cos²x-cos²x+sin²x

(√ (0,5+sin²x)) ² = (2*sin²x) ²

0,5+sinx²=4sin⁴x

4*sin⁴x-sin²x-0,5=0

sin²x=t≥0 ⇒

4t²-t-0,5=0 |*2

8t²-2t-1=0 D=36

t₁=-0,25 ∉

t₂=0,5 sin²x=0,5=1/2

sinx=+/-√ (1/2) = + / - √2/2

x₁=π/4+2πn x₂=3π/4+2πn x₃=5π/4+2πn x₄=7π/4+2πn ⇒

Ответ: x=π/4+πn/2.