Prove that: 100^ + 20^ = 40^ - Vidyadip

Question

Prove that:
$\cos100^\circ+\cos20^\circ=\cos40^\circ$

✓

Answer

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

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