Which one of the following number (s) is/are rational?

Solution:

we have,

$\sin {15}^{\circ }=\sin ({45}^{\circ }-{30}^{\circ })=\frac{\sqrt{3}-1}{2\sqrt{2}}$

$\Rightarrow \sin {15}^{\circ }=\frac{1}{4}(\sqrt{6}-\sqrt{2})$ which is an irrational number

     $\cos {15}^{\circ }=\cos ({45}^{\circ }-{30}^{\circ })=-\frac{\sqrt{3}+1}{2\sqrt{2}}$

$\Rightarrow \cos {15}^{\circ }=\frac{1}{4}(\sqrt{6}+\sqrt{2})$ which is an irrational number

       $\sin {15}^{\circ }\cos {15}^{\circ }=\frac{1}{2}\sin {30}^{\circ }=\frac{1}{4}$which is a rational number

      $sin{15}^{\circ }cos{75}^{\circ }=si{n}^2{15}^{\circ }=\frac{1}{2}(1-cos{30}^{\circ })=\frac{2-\sqrt{3}}{4}$ which is an irrational number