1. ## prove

$\displaystyle \sin 47° + \sin 61° - \sin 11° - \sin 25° = \cos 7°$.

2. ## Sum to Product

First, note that 47-11=61-25=36, so rearrange the left hand side to get $\displaystyle (\sin 47^\circ - \sin 11^\circ) + (\sin 61^\circ - \sin 25^\circ)$. From this, we use the sum-to-product formulas, specifically the formula

$\displaystyle \sin A - \sin B = 2\cos \left ( \frac{A+B}{2} \right ) \sin \left ( \frac{A-B}{2} \right )$.

Thus we get
$\displaystyle \sin 47^\circ + \sin 61^\circ - \sin 11^\circ - \sin 25^\circ = (\sin 47^\circ - \sin 11^\circ) + (\sin 61^\circ - \sin 25^\circ)$
$\displaystyle \,\,= 2 \cos 29^\circ \sin 18^\circ + 2 \cos 43^\circ \sin 18^\circ$
$\displaystyle \,\,= 2(\cos 29^\circ + \cos 43^\circ)\sin 18^\circ$

We use the sum-to-product formulas again, this time using $\displaystyle \cos A + \cos B = 2\cos \left ( \frac{A+B}{2} \right ) \cos \left ( \frac{A-B}{2} \right )$, so
$\displaystyle \sin 47^\circ + \sin 61^\circ - \sin 11^\circ - \sin 25^\circ = 2(\cos 29^\circ + \cos 43^\circ)\sin 18^\circ$
$\displaystyle \,\,= 2(2\cos 36^\circ \cos (-7^\circ))\sin 18^\circ$
$\displaystyle \,\,= 4\cos 7^\circ \cos 36^\circ \sin 18^\circ$, as $\displaystyle \cos(-x)=\cos x$.

Working from the geometry of the pentagon, one can find the sine and cosine for angles that are multiples of 18 degrees, and thus find that:
$\displaystyle \cos 36^\circ = \frac{1 + \sqrt{5}}{4}$, $\displaystyle \sin 18^\circ = \frac{-1 + \sqrt{5}}{4}$, and $\displaystyle \cos 36^\circ \sin 18^\circ = \frac{1 + \sqrt{5}}{4}\frac{-1 + \sqrt{5}}{4}=\frac{5-1}{16}=\frac{1}{4}$

So we get:
$\displaystyle \sin 47^\circ + \sin 61^\circ - \sin 11^\circ - \sin 25^\circ = 4\cos 7^\circ \cos 36^\circ \sin 18^\circ$
$\displaystyle \sin 47^\circ + \sin 61^\circ - \sin 11^\circ - \sin 25^\circ = 4\cos 7^\circ \cdot \frac{1}{4}=\cos 7^\circ$

--Kevin C.