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Q. $\cos \frac{7 \pi}{8}+\cos \frac{\pi}{4}+\cos \left(\frac{-\pi}{8}\right)-1=$

TS EAMCET 2021

Solution:

$ \cos \left(\frac{7 \pi}{8}\right)+\cos \left(\frac{\pi}{4}\right)+\cos \left(\frac{-\pi}{8}\right)-1 $
$ =\cos \left(\frac{7 \pi}{8}\right)+\cos \left(\frac{\pi}{8}\right)+\cos \left(\frac{\pi}{4}\right)-1 $
$ =2 \cos \left(\frac{\pi}{2}\right) \cos \left(\frac{3 \pi}{8}\right)+\cos \left(\frac{\pi}{4}\right)-1=\frac{1}{\sqrt{2}}-1$
Option $(a) 4 \cos \left(\frac{\pi}{16}\right) \cos \left(\frac{3 \pi}{4}\right) \cos \left(\frac{5 \pi}{8}\right) $
$ =2 \cdot 2 \cos \left(\frac{\pi}{16}\right) \cos \left(\frac{10 \pi}{16}\right) \cos \left(\frac{3 \pi}{4}\right) $
$=2 \cdot\left[\cos \left(\frac{11 \pi}{16}\right)+\cos \left(\frac{9 \pi}{16}\right)\right] \cos \left(\frac{3 \pi}{4}\right) $
$ =\cos \left(\frac{23 \pi}{16}\right)+\cos \left(\frac{\pi}{16}\right)+\cos \left(\frac{21 \pi}{16}\right)+\cos \left(\frac{3 \pi}{16}\right)$
Option (b) $ 4 \cos \left(\frac{\pi}{16}\right) \cos \left(\frac{\pi}{8}\right) \cos \left(\frac{5 \pi}{8}\right) $
$ =2 \cdot\left[\cos \left(\frac{3 \pi}{16}\right)+\cos \left(\frac{\pi}{16}\right)\right] \cos \left(\frac{5 \pi}{8}\right)$
$ =\cos \left(\frac{13 \pi}{16}\right)+\cos \left(\frac{7 \pi}{16}\right)+\cos \left(\frac{9 \pi}{16}\right)+\cos \left(\frac{11 \pi}{16}\right) $
Option (c) $ 4 \cos \left(\frac{\pi}{16}\right) \cos \left(\frac{3 \pi}{8}\right) \cos \left(\frac{9 \pi}{16}\right) $
$=2 \cdot\left[\cos \left(\frac{7 \pi}{16}\right)+\cos \left(\frac{5 \pi}{16}\right)\right] \cos \left(\frac{9 \pi}{16}\right) $
$ =\cos \left(\frac{16 \pi}{16}\right)+\cos \left(\frac{2 \pi}{16}\right)+\cos \left(\frac{4 \pi}{16}\right)+\cos \left(\frac{14 \pi}{16}\right) $
$ =-1+\cos \left(\frac{\pi}{8}\right)+\frac{1}{\sqrt{2}}+\cos \left(\frac{7 \pi}{8}\right)=-1+\frac{1}{\sqrt{2}}$