Question

$\cos {35}^{\circ }+\cos {85}^{\circ }+\cos {155}^{\circ }=$

• A. $\frac{1}{\sqrt{3}}$
  
  
• B. $\frac{1}{\sqrt{2}}$
  
• C. $\cos {275}^{\circ }$
  
• D. $0$

Answer 1

The correct option is D $0$

$\cos {35}^{\circ }+\cos {85}^{\circ }+\cos {155}^{\circ }=$

$=2\cos \left(\frac{{35}^{\circ }+{85}^{\circ }}{2}\right)\cos \left(\frac{{35}^{\circ }-{85}^{\circ }}{2}\right)+\cos {155}^{\circ }$

$[\because \cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)]$

$=2\cos {60}^{\circ }\cos (-25{)}^{\circ }+\cos {155}^{\circ }$

$=2\times \frac{1}{2}\cos {25}^{\circ }+\cos {155}^{\circ }$

$=\cos {25}^{\circ }+\cos {155}^{\circ }$

$=2\cos \left(\frac{{25}^{\circ }+{155}^{\circ }}{2}\right)\cos \left(\frac{{25}^{\circ }-{155}^{\circ }}{2}\right)$

$[\because \cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)]$

$=2\cos \left(\frac{{180}^{\circ }}{2}\right)\cos \left(\frac{-{130}^{\circ }}{2}\right)$

$=2\cos {90}^{\circ }\cos {65}^{\circ }$

$=0$

Hence, Option (A) is correct.