What is the precise value of the expression $\frac{\cos 25^{\circ}+\sin (-5^{\circ})}{\sin 25^{\circ}+\cos 5^{\circ}}$

Question

The values are in degrees. I did various manipulations of the expression but the best I could was to end up with an expression in $\sin 20^\circ$ and $\cos 20^\circ$ which are hard to compute. As this was an exam question I think it must be a quick elegant solution.

Answer 1

$$\frac{\cos25^{\circ}-\sin5^{\circ}}{\sin25^{\circ}+\cos5^{\circ}}=\frac{\sin65^{\circ}-\sin5^{\circ}}{\sin25^{\circ}+\sin85^{\circ}}=\frac{2\sin30^{\circ}\cos35^{\circ}}{2\sin55^{\circ}\cos30^{\circ}}=\tan30^{\circ}=\frac{1}{\sqrt3}$$

Answer 2

Conversion from cosines to sines gives $$\frac {\sin 65° - \sin 5°}{\sin 25°+\sin 85°}$$ $$=\frac {2.\cos 65° . \sin 30°}{2\sin 55°.\cos 30°}$$ $$=\frac {2.\cos 65° . \sin 30°}{2\cos 35°.\cos 30°}$$ $$\tan 30°=\frac {1}{\sqrt 3}$$