# Which of the following number is/are rational?

$(a)\;\sin 15^{\large\circ}\qquad(b)\;\cos 15^{\large\circ}\qquad(c)\;\sin 15^{\large\circ}\cos 15^{\large\circ}\qquad(d)\;\sin 15^{\large\circ}\cos 75^{\large\circ}$

$\sin 15^{\large\circ}=\sin(45^{\large\circ}-30^{\large\circ})$
$\qquad\quad=\large\frac{\sqrt 3-1}{2\sqrt 2}\Rightarrow$Not rational
$\cos 15^{\large\circ}=\cos(45^{\large\circ}-30^{\large\circ})$
$\qquad\quad=\large\frac{\sqrt 3+1}{2\sqrt 2}\Rightarrow$Not rational
$\sin 15^{\large\circ}\cos 15^{\large\circ}=\large\frac{1}{2}$$.2\sin 15^{\large\circ}\cos 15^{\large\circ} \qquad\qquad\qquad=\large\frac{1}{2}$$\sin 30^{\large\circ}$
$\qquad\qquad\qquad=\large\frac{1}{2}\times\frac{1}{2}$
$\qquad\qquad\qquad=\large\frac{1}{4}\Rightarrow$Rational
$\sin 15^{\large\circ}\cos 15^{\large\circ}=\sin 15^{\large\circ}(\cos 90^{\large\circ}-15^{\large\circ})$
$\qquad\qquad\qquad=\sin^215^{\large\circ}$
$\qquad\qquad\qquad=\big(\large\frac{\sqrt 3-1}{2\sqrt 2}\big)^2$
$\qquad\qquad\qquad=\large\frac{4-2\sqrt 3}{8}\Rightarrow$Not rational
Hence (c) is the correct option.