Question

${\cos }^{2}A+{\cos }^2({120}^{\circ }+A)+co{s}^2({120}^{\circ }-A)=$

• A. $\frac{3}{2}$
• B. $\frac{1}{2}$
• C. $\frac{3}{18}$
• D. $\frac{1}{4}$

Answer 1

The correct option is A $\frac{3}{2}$

${\cos }^{2}A+{\cos }^2({120}^{\circ }+A)+{\cos }^2({120}^{\circ }-A)={\cos }^{2}A+\frac{1}{2}(1+\cos ({240}^{\circ }+2A\left)\right)+\frac{1}{2}(1+\cos ({240}^{\circ }-2A\left)\right)$

$={\cos }^{2}A+1+\frac{1}{2}\left(\cos \right({240}^{\circ }+2A)+\cos ({240}^{\circ }-2A\left)\right)$

$=1+{\cos }^{2}A+\frac{1}{2}\left(2\cos {240}^{\circ }\cos 2A\right)$

$=1+{\cos }^{2}A-\frac{1}{2}(2{\cos }^{2}A-1)$

$=1+\frac{1}{2}=\frac{3}{2}$