Solve-for-real-numbers-sinx-1-sin-2-x-1-cosy-1-cos-2-y-

Question Number 184622 by Shrinava last updated on 09/Jan/23

Solve for real numbers :

sinx \sqrt{1 - \sin^{2} x} = 1 + cosy \sqrt{1 - \cos^{2} y}

Answered by floor(10²Eta[1]) last updated on 09/Jan/23

sinxcosx = 1 + senycosy

\frac{sen (2 x)}{2} = 1 + \frac{sen (2 y)}{2}

sen (2 x) = 2 + sen (2 y)

- 1 ⩽ sen (2 x) ⩽ 1

1 ⩽ 2 + sen (2 y) ⩽ 3

\Rightarrow sen 2 x = 1 = 2 + sen 2 y

\Rightarrow sen 2 x = 1 \land sen 2 y = - 1

2 x = \frac{π}{2} + 2 n π \land 2 y = \frac{3 π}{2} + 2 m π

\Rightarrow x = \frac{π}{4} + n π \land y = \frac{3 π}{4} + m π, m, n \in Z