WLOG (without loss of generality) let the length of the hypotenuse be equal to 1.  

Can you find two different expressions for $a$?  Can you find two different expressions for $b$?

Can you find a relationship connecting $\sin^2(45^{\circ} +x)$ and $\sin^2(45^{\circ}-x)$?

You can draw the graph of the sine function. What about the graph of $y = \sin^2 x$? What about the periodicity of this graph?