Prove that: 55^ + 65^ 175^ =0 - Vidyadip

Question

Prove that:
$\cos55^\circ+\cos65^\circ\cos175^\circ=0$

✓

Answer

Consider LHS:
$\cos55^\circ+\cos65^\circ+\cos175^\circ$
$=\ 2\cos\Big(\frac{55^\circ+65^\circ}{2}\Big)\cos\Big(\frac{55^\circ-65^\circ}{2}\Big)+\cos175^\circ$ $\Big\{\because\ \cos\text{A}+\cos\text{B}=2\cos\Big(\frac{\text{A+B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)\Big\}$
$=\ 2\cos60^\circ\cos(-5^\circ)+\cos175^\circ$
$=\ 2\times\frac{1}{2}\cos5^\circ+\cos175^\circ$
$=\ \cos5^\circ+\cos175^\circ$
$=\ 2\cos\Big(\frac{5^\circ+175^\circ}{2}\Big)\cos\Big(\frac{5^\circ-175^\circ}{2}\Big)$
$=\ 2\cos90^\circ\cos85^\circ$
$=\ 0=\text{RHS}$
Hence, LHS = RHS.

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