The answer is 2.
METHOD
3π/8=π-5π/8. cos(π-5π/8)=-cos(5π/8)=cos(3π/8).
sin(π-5π/8)=sin(5π/8)=sin(3π/8).
Therefore sin²(5π/8)+cos²(3π/8)=sin²(5π/8)+cos²(5π/8)=1.
Similarly, 17π/18=3π-37π/18. cos(3π-37π/18)=-cos(37π/18)=cos(17π/8)
So sin²(17π/18)+cos²(37π/18)=1
Total sum is 1+1=2.