Question

Evaluate: $\cos {20}^{\circ }\cos {100}^{\circ }+\cos {100}^{\circ }\cos {140}^{\circ }-\cos {140}^{\circ }\cos {200}^{\circ }$

• A. $\frac{3}{4}$
• B. $\frac{1}{4}$
• C. $\frac{-1}{4}$
• D. $\frac{-3}{4}$

Answer 1

The correct option is D $\frac{-3}{4}$

LHS$=\cos {20}^{\circ }\cos {100}^{\circ }+\cos {100}^{\circ }\cos {140}^{\circ }-\cos {140}^{\circ }\cos {200}^{\circ }$

$=\frac{1}{2}(2\cos {20}^{\circ }\cos {100}^{\circ }+2\cos {100}^{\circ }\cos {140}^{\circ }-2\cos {140}^{\circ }\cos {200}^{\circ })$

$=\frac{1}{2}[\cos {120}^{\circ }+\cos {80}^{\circ }+\cos {240}^{\circ }+\cos {40}^{\circ }-\cos {340}^{\circ }-\cos {60}^{\circ }]$

$=\frac{1}{2}[\cos ({180}^{\circ }-{60}^{\circ })+\cos {80}^{\circ }+\cos ({270}^{\circ }-{30}^{\circ })+\cos {40}^{\circ }-\cos ({360}^{\circ }-{20}^{\circ })-\cos {60}^{\circ }]$

$=\frac{1}{2}[-\frac{1}{2}+\cos {80}^{\circ }-\frac{1}{2}+\cos {40}^{\circ }-\cos ({360}^{\circ }-{20}^{\circ })-\frac{1}{2}]$

$=\frac{1}{2}[-\frac{3}{2}+\cos {80}^{\circ }+\cos {40}^{\circ }-\cos {20}^{\circ }]$

$=\frac{1}{2}[-\frac{3}{2}+2\cos {60}^{\circ }\cos {20}^{\circ }-\cos {20}^{\circ }]$

$=\frac{1}{2}[-\frac{3}{2}+2\times \frac{1}{2}\cos {20}^{\circ }-\cos {20}^{\circ }]$

$=\frac{1}{2}[-\frac{3}{2}+\cos {20}^{\circ }-\cos {20}^{\circ }]$

$=-\frac{3}{4}$