Rational fools

$A) \ \cos 30^\circ = 1 - 2\sin^2 15^\circ \\ \Rightarrow \dfrac{\sqrt{3}}{2}= 1 - 2\sin^2 15^\circ \\ \Rightarrow \sqrt{3} = 2 - 4\sin^2 15^\circ \\ \Rightarrow \sin^2 15^\circ = \dfrac{2-\sqrt{3}}{4} \Rightarrow \sin 15^\circ \ is \ irrational$

$B) \sin^2 15^\circ = \dfrac{2-\sqrt{3}}{4} \\ \Rightarrow \cos^2 15^\circ = 1-\dfrac{2-\sqrt{3}}{4} \\ \Rightarrow \cos 15^\circ \ is \ irrational$

$C) 2\sin 15^\circ \cos 15^\circ =\sin 30^\circ = \dfrac{1}{2} \\ \Rightarrow \sin 15^\circ \cos 15^\circ \ is \ rational$

$D) \sin 15^\circ \cos 75^\circ = \sin 15^\circ \sin 15^\circ \\ =\sin^2 15^\circ = \dfrac{2-\sqrt{3}}{4} \\ \Rightarrow \sin 15^\circ \cos 75^\circ \ is \ irrational$

Thus , only $C$ is rational.

$A\quad =\quad sin\quad 15\quad :\quad \\ A=\quad \frac { \sqrt { 6 } -\sqrt { 2 } }{ 4 } \\ B=\quad cos\quad 15\quad \Longrightarrow \frac { \sqrt { 6 } +\sqrt { 2 } }{ 4 } \\ C=\quad (sin15)*(cos15)\quad \Longrightarrow \frac { 1 }{ 4 } \\ D=(sin15)*(cos75)\Longrightarrow \frac { 2-\sqrt { 3 } }{ 4 } \\ \\ \therefore \quad only\quad C\quad is\quad rational\quad ,\quad all\quad other\quad three\quad of\quad them\quad are\quad irrational\quad$