# Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 227: 25

$$\sin76^\circ\cos31^\circ-\cos76^\circ\sin 31^\circ=\frac{\sqrt2}{2}$$

#### 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, $$\sin76^\circ\cos31^\circ-\cos76^\circ\sin 31^\circ=\frac{\sqrt2}{2}$$

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