\(A=cos10^o.cos50^o.cos70^o=\dfrac{cos10^o}{2}\left[cos120^o+cos20^o\right]=\dfrac{cos10^o}{2}\left[cos20^o-\dfrac{1}{2}\right]\)
\(=\dfrac{1}{2}\left[cos10^o.cos20^o-\dfrac{1}{2}cos10^o\right]\)
\(=\dfrac{1}{2}\left[\dfrac{1}{2}\left(cos30^o+cos10^o\right)-\dfrac{1}{2}cos10^o\right]=\dfrac{1}{4}.\dfrac{\sqrt{3}}{2}=\dfrac{\sqrt{3}}{8}\)
\(B=\left(cos\dfrac{\pi}{5}+cos\dfrac{4\pi}{5}\right)+\left(cos\dfrac{2\pi}{5}+cos\dfrac{3\pi}{5}\right)\)
\(=2cos\dfrac{\pi}{2}.cos\dfrac{3\pi}{10}+2cos\dfrac{\pi}{2}.cos\dfrac{\pi}{10}=0\)
\(C=cos\dfrac{2\pi}{7}+cos\dfrac{4\pi}{7}+cos\dfrac{6\pi}{7}\)
\(\Rightarrow C.sin\dfrac{\pi}{7}=sin\dfrac{\pi}{7}.cos\dfrac{2\pi}{7}+sin\dfrac{\pi}{7}.cos\dfrac{4\pi}{7}+sin\dfrac{\pi}{7}.cos\dfrac{6\pi}{7}\)
\(=\dfrac{1}{2}\left[sin\dfrac{3\pi}{7}+sin\dfrac{-\pi}{7}+sin\dfrac{5\pi}{7}+sin\dfrac{-3\pi}{7}+sin\pi+sin\dfrac{-5\pi}{7}\right]\)
\(=\dfrac{1}{2}\left(sin\pi-sin\dfrac{\pi}{7}\right)=-\dfrac{1}{2}sin\dfrac{\pi}{7}\) \(\Rightarrow C=-\dfrac{1}{2}\)
