Question

The value of $(\cot {18}^{\circ }\cot {12}^{\circ }-\cot {45}^{\circ })({\sin }^2{15}^{\circ }-{\sin }^2{3}^{\circ })$ is

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

Answer 1

The correct option is B $\frac{\sqrt{3}}{2}$

$(\cot {18}^{\circ }\cot {12}^{\circ }-\cot {45}^{\circ })({\sin }^2{15}^{\circ }-{\sin }^2{3}^{\circ })=(\frac{\cos {18}^{\circ }\cos {12}^{\circ }}{\sin {18}^{\circ }\sin {12}^{\circ }}-1)(\sin (15+3{)}^{\circ }\cdot \sin (15-3{)}^{\circ })=\left(\frac{\cos {18}^{\circ }\cos {12}^{\circ }-\sin {18}^{\circ }\sin {12}^{\circ }}{\sin {18}^{\circ }\sin {12}^{\circ }}\right)(\sin {18}^{\circ }\cdot \sin {12}^{\circ })=\left(\frac{\cos ({18}^{\circ }+{12}^{\circ })}{\sin {18}^{\circ }\sin {12}^{\circ }}\right)(\sin {18}^{\circ }\cdot \sin {12}^{\circ })=\cos {30}^{\circ }=\frac{\sqrt{3}}{2}$