Question

The value of $\frac{\sin 70°+\cos 40°}{\cos 70°+\sin 40°}$ is _________.

Answer 1

​$$\begin{gathered} \frac{\sin 70°+\cos 40°}{\cos 70°+\sin 40°} \\ =\frac{\sin 70°+\cos \left(90°-50°\right)}{\cos \left(90°-20°\right)+\sin 40°}\left[\because \cos \left(90°-\theta \right)-\sin \theta \right] \\ =\frac{\sin 70°+\sin 50°}{\sin 20°+\sin 40°} \\ =\frac{2\sin \left({\displaystyle \frac{70°+50°}{2}}\right)\cos \left({\displaystyle \frac{70°-50°}{2}}\right)}{2\sin \left({\displaystyle \frac{20°+40°}{2}}\right)\cos \left({\displaystyle \frac{40°-20°}{2}}\right)}\left[\because \sin A+\sin B=2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)\right] \\ =\frac{2\sin 60°\cos 10°}{2\sin 30°\cos 10°} \\ =\frac{{\displaystyle \frac{\sqrt{3}}{2}}}{{\displaystyle \frac{1}{2}}} \\ =\sqrt{3} \\ \mathrm{Hence},\frac{\sin 70°+\cos 40°}{\cos 70°+\sin 40°}=\sqrt{3} \end{gathered}$$