Question

The value of ${\cos }^2{75}^o+{\cos }^2{45}^o+{\cos }^2{15}^o-{\cos }^2{30}^o-{\cos }^2{60}^o$ is

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{1}{2}$

Given ${\cos }^2{75}^o+{\cos }^2{45}^o+{\cos }^2{15}^o-{\cos }^2{30}^o-{\cos }^2{60}^o$
${(\frac{\sqrt{3}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}})}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+{(\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}})}^2-{\left(\frac{\sqrt{3}}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2$
$=\left(\frac{3+1-2\sqrt{3}}{8}\right)+\frac{1}{2}+\left(\frac{3+1+2\sqrt{3}}{8}\right)-\frac{3}{4}-\frac{1}{4}$
$=\frac{1}{2}-\frac{\sqrt{3}}{4}+\frac{1}{2}+\frac{1}{2}+\frac{\sqrt{3}}{4}-\frac{3}{4}-\frac{1}{4}-1=\frac{1}{2}$