Question

The value of the expression $\cos {20}^{\circ }\cos {100}^{\circ }+\cos {100}^{\circ }\cos {140}^{\circ }-\cos {140}^{\circ }\cos {200}^{\circ }$ is

• A. $-\frac{1}{2}$
• B. $-\frac{3}{4}$
• C. $1$
• D. $-\frac{1}{2}$

Answer 1

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

$\cos {20}^{\circ }\cos {100}^{\circ }+\cos {100}^{\circ }\cos {140}^{\circ }-\cos {140}^{\circ }\cos {200}^{\circ }=\cos {100}^{\circ }(\cos {20}^{\circ }+\cos {140}^{\circ })-\cos {140}^{\circ }\cos {200}^{\circ }$
$=2\cos {100}^{\circ }\cos {80}^{\circ }\cos {60}^{\circ }-\cos {140}^{\circ }\cos {200}^{\circ }$
$=\cos {100}^{\circ }\cos {80}^{\circ }-\cos {140}^{\circ }\cos {200}^{\circ }$
$=\left(\frac{\cos {180}^{\circ }+\cos {20}^{\circ }}{2}\right)-\left(\frac{\cos {340}^{\circ }+\cos {60}^{\circ }}{2}\right)$
$=\left(\frac{-1+\cos {20}^{\circ }}{2}\right)-\left(\frac{\cos {20}^{\circ }+\frac{1}{2}}{2}\right)=-\frac{1}{2}+\frac{\cos {20}^{\circ }}{2}-\frac{\cos {20}^{\circ }}{2}-\frac{1}{4}$
$=-\frac{1}{2}-\frac{1}{4}$
$=-\frac{3}{4}$