Discussion
The value of \(\frac{\sqrt{3}(sin40^{o} + sin20^{o})}{cos40^{o} + cos20^{o}}\) is
Answer: A
The value is 1.
Use the sum-to-product formulas on the numerator and denominator.
\[ \begin{aligned} \sin 40^\circ + \sin 20^\circ &= 2\sin 30^\circ\cos 10^\circ \\ \cos 40^\circ + \cos 20^\circ &= 2\cos 30^\circ\cos 10^\circ \\ \text{value} &= \sqrt{3}\tan 30^\circ = \sqrt{3} \times \tfrac{1}{\sqrt{3}} = 1 \end{aligned} \]
Forgetting the factor \(\sqrt{3}\) in front gives \(\frac{1}{\sqrt{3}}\), and \(\sqrt{3}\) itself is \(\tan 60^\circ\), the wrong half-sum.
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