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If $\cot \theta + \cot \left( {\frac{\pi }{4} + \theta } \right) = 2$, then the general value of $\theta $ is
$2n\pi \pm \frac{\pi }{6}$
$2n\pi \pm \frac{\pi }{3}$
$n\pi \pm \frac{\pi }{3}$
$n\pi \pm \frac{\pi }{6}$
Solution
(d) $\cot \theta + \cot \left( {\frac{\pi }{4} + \theta } \right) = 2$
$ \Rightarrow \frac{{\cos \theta }}{{\sin \theta }} + \frac{{\cos \{ (\pi /4) + \theta \} }}{{\sin \{ (\pi /4) + \theta \} }} = 2$
$ \Rightarrow $ $\sin \left( {\frac{\pi }{4} + 2\theta } \right) = 2\sin \theta \sin \left( {\frac{\pi }{4} + \theta } \right)$
$ \Rightarrow $ $\sin \left( {\frac{\pi }{4} + 2\theta } \right) + \cos \left( {\frac{\pi }{4} + 2\theta } \right) = \frac{1}{{\sqrt 2 }}$
$ \Rightarrow $ $\cos 2\theta = \frac{1}{2} $
$\Rightarrow 2\theta = 2n\pi \pm \frac{\pi }{3} $
$\Rightarrow \theta = n\pi \pm \frac{\pi }{6}$.
