Let and be non-zero real numbers such that 2 ( cos - cos ) + cos cos = 1 . Then which of the following is/are true? [2017]

Question

Let $α$ and $β$ be non-zero real numbers such that $2 (\cos β - \cos α) + \cos α \cos β = 1$ . Then which of the following is/are true? [2017]

Smart Achievers · Accepted Answer

(1, 3)

If we consider $β$ and $\tan \frac{β}{2} = y,$ then

$2 (\cos β - \cos α) + \cos α \cos β = 1$

$\Rightarrow 2 [\frac{1 - y^{2}}{1 + y^{2}} - \frac{1 - x^{2}}{1 + x^{2}}] = 1 - \frac{(1 - x^{2}) (1 - y^{2})}{(1 + x^{2}) (1 + y^{2})}$

$\Rightarrow 2 [(1 + x^{2}) (1 - y^{2}) - (1 - x^{2}) (1 + y^{2})] = (1 + x^{2}) (1 + y^{2}) - (1 - x^{2}) (1 - y^{2})$

$\Rightarrow 4 (x^{2} - y^{2}) = 2 (x^{2} + y^{2})$

$\Rightarrow x^{2} = 3 y^{2} \Rightarrow x = \pm \sqrt{3} y$

$\Rightarrow \tan \frac{α}{2} \pm \sqrt{3} \tan \frac{β}{2} = 0$

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