If `x=sin(theta+(7pi)/(12))+sin(theta+pi/(12))+sin(theta+(3pi)/(12))dot` `Y=cos(theta+(7pi)/(12))+cos(theta+pi/(12))+cos(theta+(3pi)/(12))dot` then prove that `X/Y-Y/X=2tan2theta`

Explanation

To prove that \( \frac{X}{Y} - \frac{Y}{X} = 2 \tan 2\theta \), we start with the definitions of \( X \) and \( Y \):

\[ X = \sin\left(\theta + \frac{7\pi}{12}\right) + \sin\left(\theta + \frac{\pi}{12}\right) + \sin\left(\theta + \frac{3\pi}{12}\right) \]

\[
Y = \cos\left(\theta + \frac{7\pi}{12}\right) + \cos\left(\theta + \frac{\pi}{12}\right) + \cos\left(\theta + \frac{3\pi}{12}\right)
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