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Trigonometric transformations

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Question

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

Easy

Solution

$\begin{array}{l} \frac{2 \sin 4^{\circ} \cos 3^{\circ} + 2 \sin 4^{\circ} \cos 1^{\circ}}{\cos 1^{\circ} \cos 2^{\circ} \sin 4^{\circ}} \\ = \frac{2 \sin 4^{\circ} [\cos 3^{\circ} + \cos 1^{\circ}]}{\cos 1^{\circ} \cos 2^{\circ} \sin 4^{\circ}} \\ = \frac{4 \cos 2^{\circ} \cos 1^{\circ}}{\cos 1^{\circ} \cos 2^{\circ}} = 4 \end{array}$

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