Question

cos 20+cos80-√3cos50=?

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{cos\,20^\circ+cos\,80^\circ-\sqrt{3}\;cos\,50}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}$

$\mathsf{cos\,20^\circ+cos\,80^\circ-\sqrt{3}\;cos\,50}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Idenity used:}}$

$\boxed{\mathsf{cos\,C+cos\,D=2\,cos\left(\dfrac{C+D}{2}\right)\,cos\left(\dfrac{C-D}{2}\right)}}$

$\mathsf{Consider,}$

$\mathsf{cos\,20^\circ+cos\,80^\circ-\sqrt{3}\;cos\,50}$

$\textsf{Using the above identity, we get}$

$\mathsf{=2\,cos\left(\dfrac{20^\circ+80^\circ}{2}\right)\,\,cos\left(\dfrac{20^\circ-80^\circ}{2}\right)-\sqrt{3}\;cos\,50}$

$\mathsf{=2\,cos\,50^\circ\,\,cos(-30^\circ)-\sqrt{3}\;cos\,50}$

$\mathsf{=2\,cos\,50^\circ\,\,cos\,30^\circ-\sqrt{3}\;cos\,50}$

$\mathsf{=2\,cos\,50^\circ\left(\dfrac{\sqrt3}{2}\right)-\sqrt{3}\;cos\,50}$

$\mathsf{=\sqrt{3}\,cos\,50^\circ-\sqrt{3}\;cos\,50}$

$\mathsf{=0}$

$\implies\boxed{\mathsf{cos\,20^\circ+cos\,80^\circ-\sqrt{3}\;cos\,50=0}}$