Prove that: 20^ + 100^ + 140^ =0 - Vidyadip

Question

Prove that:
$\cos20^\circ+\cos100^\circ+\cos140^\circ=0$

✓

Answer

$\text{LHS}=\cos20^\circ+\cos100^\circ+\cos140^\circ$
$\Rightarrow\ (\cos20^\circ+\cos100^\circ)+\cos140^\circ$
$=\ 2\cos\Big(\frac{20^\circ+100^\circ}{2}\Big)\cos\Big(\frac{20^\circ-100^\circ}{2}\Big)+\cos140^\circ$ $\Big[\because\ \cos\text{c}-\cos\text{D}=2\cos\Big(\frac{\text{C}-\text{D}}{2}\Big)\cos\Big(\frac{\text{C}-\text{D}}{2}\Big)\Big]$
$=\ 2\cos60^\circ\cos(-40^\circ)+\cos140^\circ$
$=\ 2\times\frac{1}{2}\cos40+\cos140^\circ$ $\Big[\because\ \cos60^\circ=\frac{1}{2}\Big]$
$=\ \cos40^\circ+\cos(180^\circ-40^\circ)$
$=\ \cos40^\circ-\cos40^\circ$
$=\ 0$
$=\ \text{RHS}$

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