Question

The value of ${\cos }^2{45}^{\circ }-{\sin }^2{15}^{\circ }$ is:

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

Answer 1

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

${\cos }^2{45}^{\circ }-{\sin }^2{15}^{\circ }$
We know that ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$
$\Rightarrow {\cos }^2{45}^{\circ }-{\sin }^2{15}^{\circ }={\left(\frac{1}{\sqrt{2}}\right)}^2-\frac{1-\cos {30}^{\circ }}{2}$
$=\frac{1-1+\frac{\sqrt{3}}{2}}{2}$
$=\frac{\sqrt{3}}{4}$