We have the following:
We have that the cosine and the sine are equal their quotient, that is, tangent, is 1. And this happens when the angle is 45 degrees.
[tex]\begin{gathered} \sin X=-\cos Y \\ \tan \frac{X}{Y}=1 \end{gathered}[/tex]We know that the second quadrant (between 90 and 180 degrees), the sine is negative and the cosine is positive, therefore, in 135 (90 + 45) degree it will be -1.
Therefore the answer is:
[tex]\begin{gathered} X=135 \\ Y=135 \\ X+Y=135+135=270 \end{gathered}[/tex]