Question

The value of $\frac{\sin 1^{\circ }+\sin 3^{\circ }+\sin 5^{\circ }+\sin 7^{\circ }}{\cos 1^{\circ }\cos 2^{\circ }\sin 4^{\circ }}$ is

Answer 1

Given:

$\frac{\sin 1^{\circ }+\sin 3^{\circ }+sin{5}^{\circ }+sin{7}^{\circ }}{\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \sin 4^{\circ }}$

$=\frac{(\sin 1^{\circ }+\sin 5^{\circ })+(sin{3}^{\circ }+sin{7}^{\circ })}{\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \sin 4^{\circ }}$
$=\frac{2\sin 3^{\circ }\cos 2^{\circ }+2\sin 5^{\circ }\cos 2^{\circ }}{\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \sin 4^{\circ }}$
$=\frac{2\cos 2^{\circ }(\sin 3^{\circ }+\sin 5^{\circ })}{\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \sin 4^{\circ }}$
$=\frac{4\cos 2^{\circ }\cos 1^{\circ }\sin 4^{\circ }}{\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \sin 4^{\circ }}$
$=4$