Prove that: 80^ - 70^ = 50^ - Vidyadip

Question

Prove that:
$\sin80^\circ-\cos70^\circ=\cos50^\circ$

✓

Answer

$\sin80^\circ-\cos70^\circ=\cos50^\circ$
$\text{LHS}=\sin80^\circ=\cos50^\circ+\cos70^\circ$
Now,
$\cos\text{C}+\cos\text{D}=2\cos\frac{\text{C+D}}{2}\cos\frac{\text{C}-\text{D}}{2}$
$\text{RHS}=\cos50^\circ+\cos70^\circ$
$=2\cos\Big(\frac{50^\circ+70^\circ}{2}\Big)\cos\Big(\frac{50^\circ-70^\circ}{2}\Big)$
$=\ 2\cos60^\circ\cos(-10^\circ)$
$=\ 2\times\frac{1}{2}\cos10^\circ$ $[\cos(-\theta)=\cos\theta]$
$=\ \cos10^\circ$
$=\ \sin80^\circ$
$=\ \text{LHS}$ $[\because\ \cos\theta=\sin(90-\theta)]$

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