Question

The value of
${\cos }^2{10}^{\circ }-\cos {10}^{\circ }\cos {50}^{\circ }+{\cos }^2{50}^{\circ }$ is:

• A. $\frac{3}{4}+\cos {20}^{\circ }$
• B. $\frac{3}{4}$
• C. $\frac{3}{2}(1+\cos {20}^{\circ })$
• D. $\frac{3}{2}$

Answer 1

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

${\cos }^2{10}^{\circ }-\cos {10}^{\circ }\cos {50}^{\circ }+{\cos }^2{50}^{\circ }=\frac{1}{2}[2{\cos }^2{10}^{\circ }-2\cos {10}^{\circ }\cos {50}^{\circ }+2{\cos }^2{50}^{\circ }]=\frac{1}{2}[1+\cos {20}^{\circ }+1+\cos {100}^{\circ }-(\cos {60}^{\circ }+\cos {40}^{\circ }\left)\right]=\frac{1}{2}[2-\cos {60}^{\circ }+\cos {20}^{\circ }+\cos {100}^{\circ }-\cos {40}^{\circ }]=\frac{1}{2}[\frac{3}{2}+\cos {20}^{\circ }+2\sin {70}^{\circ }\sin (-{30}^{\circ }\left)\right]=\frac{1}{2}[\frac{3}{2}+\cos {20}^{\circ }-\sin {70}^{\circ }]=\frac{1}{2}\times \frac{3}{2}=\frac{3}{4}$