Answer
$$\sin76^\circ\cos31^\circ-\cos76^\circ\sin 31^\circ=\frac{\sqrt2}{2}$$
Work Step by Step
$$X=\sin76^\circ\cos31^\circ-\cos76^\circ\sin 31^\circ$$
From the identity of the difference of sines:
$$\sin A\cos B-\sin B\cos A=\sin(A-B)$$
So here $X$ actually follows the above identity with $A=76^\circ$ and $B=31^\circ$.
Therefore, X can also be rewritten as
$$X=\sin(76^\circ-31^\circ)$$
$$X=\sin45^\circ$$
$$X=\frac{\sqrt2}{2}$$
In conclusion, $$\sin76^\circ\cos31^\circ-\cos76^\circ\sin 31^\circ=\frac{\sqrt2}{2}$$
