First slide

Trigonometric transformations

Question

Let $α and β$ be non-zero real numbers such that $2 (\cos β - \cos α) + \cos αcos β = 1$ . Then which of the following is/are true?

Moderate

A

$\tan (\frac{α}{2}) + \sqrt{3} \tan (\frac{β}{2}) = 0$
B

$\sqrt{3} \tan (\frac{α}{2}) + \tan (\frac{β}{2}) = 0$
C

$\tan (\frac{α}{2}) - \sqrt{3} \tan (\frac{β}{2}) = 0$
D

$\sqrt{3} \tan (\frac{α}{2}) - \tan (\frac{β}{2}) = 0$

Solution

$\cos β (2 + \cos α) = 1 + 2 \cos α$
$\begin{array}{l} \Rightarrow \frac{\cos β}{1} = \frac{1 + 2 \cos α}{2 + \cos α} \\ \Rightarrow \frac{1 - \cos β}{1 + \cos β} = \frac{1 - \cos α}{3 (1 + \cos α)} \\ \Rightarrow \tan^{2} \frac{α}{2} = 3 \tan^{2} \frac{β}{2} \\ \Rightarrow \tan \frac{α}{2} = \pm \sqrt{3} \tan \frac{β}{2} \end{array}$

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