Question

The value of $(\cos {75}^{\circ }-\cos {15}^{\circ }{)}^2+(\sin {75}^{\circ }-\sin {15}^{\circ }{)}^2$ is

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. $-1$

Answer 1

The correct option is B $1$

$(\cos {75}^{\circ }-\cos {15}^{\circ }{)}^2+(\sin {75}^{\circ }-\sin {15}^{\circ }{)}^2$
$={\cos }^2{75}^{\circ }+{\cos }^2{15}^{\circ }-2\cos {75}^{\circ }\cos {15}^{\circ }+{\sin }^2{75}^{\circ }+{\sin }^2{15}^{\circ }-2\sin {75}^{\circ }\sin {15}^{\circ }$
$=2-2(\cos {75}^{\circ }\cos {15}^{\circ }+\sin {75}^{\circ }\sin {15}^{\circ })$
$=2(1-\cos ({75}^{\circ }-{15}^{\circ }\left)\right)=2(1-\cos {60}^{\circ })=2\times \frac{1}{2}=1$