Question

Prove that $\cos {55}^0+\cos {65}^0+\cos {175}^0=0$

Answer 1

$L.H.S.$

$\cos {55}^0+\cos {65}^0+\cos {175}^0$

We know that $\cos C+\cos D=2\cos \frac{C+D}{2}\cos \frac{C-D}{2}$

Then,

$=2\cos \frac{{55}^0+{65}^0}{2}\cos \frac{{55}^0-{65}^0}{2}+\cos {175}^0$

$=2\cos \frac{{120}^0}{2}\cos \frac{(-{10}^0)}{2}+\cos {175}^0$

$=2\cos {60}^0\cos 5^0+\cos {175}^0$ since $\cos (-\theta )=\cos \left(\theta \right)$

$=2\times \frac{1}{2}\cos 5^0+\cos {175}^0$

$=\cos 5^0+\cos {175}^0$

$\cos 5^0+\cos ({180}^0-5^0)$

$\cos 5^0-\cos 5^0$ since $\cos ({180}^0-\theta )=-\cos \theta$

$=0$

$R.H.S.$