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  4. Let k= tan -1 ( sin (π/18)+ sin (2 π/18)/ cos (π)18+ cos (2 π)18 then k equals

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Q. Let $k=\tan ^{-1} \frac{\sin \frac{\pi}{18}+\sin \frac{2 \pi}{18}}{\cos \frac{\pi}{18}+\cos \frac{2 \pi}{18}}$ then $k$ equals

NTA AbhyasNTA Abhyas 2022

Solution:

$ \theta =\tan ^{-1} \frac{\sin \frac{\pi}{18}+\sin \frac{2 \pi}{18}}{\cos \frac{\pi}{18}+\cos \frac{2 \pi}{18}} $
$=\tan ^{-1} \frac{2 \sin \frac{3 \pi}{36} \cos \frac{\pi}{36}}{2 \cos \frac{3 \pi}{36} \cos \frac{\pi}{36}} $
$=\tan ^{-1} \tan \frac{3 \pi}{36} $
$ \theta =\frac{3 \pi}{36} \text { radian }=\frac{\pi}{12}$