Question

Find the value of $\cos \dfrac{\pi }{7} + \cos \dfrac{{3\pi }}{7} + \cos \dfrac{{5\pi }}{7}$

Answer 1

In this question we will use $\cos A + \cos B = 2\cos \left( {\dfrac{{A + B}}{2}} \right)\cos \left( {\dfrac{{A - B}}{2}} \right)$ and convert $\cos \left( {\dfrac{{3\pi }}{7}} \right) + \cos \left( {\dfrac{{5\pi }}{7}} \right) = 2\cos \dfrac{{4\pi }}{7}\cos \dfrac{\pi }{7}$ then we will take $\cos \dfrac{\pi }{7}$ common and after simplifying we get the answer.

Complete step-by-step answer:
$\cos \dfrac{\pi }{7} + \cos \dfrac{{3\pi }}{7} + \cos \dfrac{{5\pi }}{7}$
We know that $\cos A + \cos B = 2\cos \left( {\dfrac{{A + B}}{2}} \right)\cos \left( {\dfrac{{A - B}}{2}} \right)$
Converting $\cos \left( {\dfrac{{3\pi }}{7}} \right) + \cos \left( {\dfrac{{5\pi }}{7}} \right)$ we get
$\Rightarrow \cos \dfrac{\pi }{7} + 2\cos \left( {\dfrac{{\dfrac{{3\pi }}{7} + \dfrac{{5\pi }}{7}}}{2}} \right)\cos \left( {\dfrac{{\dfrac{{3\pi }}{7} - \dfrac{{5\pi }}{7}}}{2}} \right)$
On simplifying we get
$\Rightarrow \cos \dfrac{\pi }{7} + 2\cos \dfrac{{4\pi }}{7}\cos \dfrac{\pi }{7}$
Taking $\cos \dfrac{\pi }{7}$ common we get
$\Rightarrow \cos \dfrac{\pi }{7}\left( {1 + 2\cos \dfrac{{4\pi }}{7}} \right)$
$\therefore \cos \dfrac{\pi }{7} + \cos \dfrac{{3\pi }}{7} + \cos \dfrac{{5\pi }}{7} = \cos \dfrac{\pi }{7}\left( {1 + 2\cos \dfrac{{4\pi }}{7}} \right)$

Note: We can also simplify it further and convert $\cos \dfrac{{4\pi }}{7} = \cos \left( {\pi - \dfrac{{3\pi }}{7}} \right) = - \cos \dfrac{{3\pi }}{7}$ then answer will become $\therefore = \cos \dfrac{\pi }{7}\left( {1 - 2\cos \dfrac{{3\pi }}{7}} \right)$ or substitute the value of $\cos \dfrac{\pi }{7}$ but it is not know so we can leave it as usual.