If $\cos x + \cos y = \frac{4}{5}$ , $\cos x - \cos y = \frac{2}{7}$ , then $14 \tan (\frac{x - y}{2}) + 5 \cot (\frac{x + y}{2}) =$

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$\frac{1}{4}$

$\frac{3}{4}$

$\frac{5}{44}$

answer is A.

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Detailed Solution

$\cos x + \cos y = \frac{4}{5}$ ---------(1)

$2 \cos (\frac{x + y}{2}) \cos (\frac{x + y}{2}) = \frac{4}{5}$

$\cos (\frac{x + y}{2}) \cos (\frac{x + y}{2}) = \frac{2}{5}$

$\cos x - \cos y = \frac{2}{7}$ -----------(2)

$- 2 \sin (\frac{x + y}{2}) \sin (\frac{x - y}{2}) = \frac{2}{7}$

$- \sin (\frac{x + y}{2}) \sin (\frac{x - y}{2}) = \frac{1}{7}$

$D i v i d i n g (2) b y (1), w e g e t$

$- \frac{\sin (\frac{x + y}{2}) \sin (\frac{x - y}{2})}{\cos (\frac{x + y}{2}) \cos (\frac{x - y}{2})} = \frac{5}{14}$

$\Rightarrow 14 \tan (\frac{x + y}{2}) = - 5 \cot (\frac{x - y}{2})$

$\Rightarrow 14 \tan (\frac{x + y}{2}) + 5 \cot (\frac{x - y}{2}) = 0$

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