Question

Which of the following numbers is rational ?

• A. $sin{15}^{\circ }+cos{15}^{\circ }$
• B. $tan22{\frac{1}{2}}^{\circ }+cot22{\frac{1}{2}}^{\circ }$
• C. $sin{15}^{\circ }-cos{15}^{\circ }$
• D. $tan22{\frac{1}{2}}^{\circ }-cot22{\frac{1}{2}}^{\circ }$

Answer 1

Answer: (D)

$tan22{\frac{1}{2}}^{\circ }-cot22{\frac{1}{2}}^{\circ }$

$formulaused$
$\sin {15}^0=\frac{\sqrt{3}-1}{2\sqrt{2}}$
$\cos {15}^0=\frac{\sqrt{2}+\sqrt{3}}{2}$
$\tan 22{\frac{1}{2}}^0=\sqrt{2}-1$
$a)\sin {15}^0+\cos {15}^0$
$=\frac{\sqrt{3}-1}{2\sqrt{2}}+\frac{\sqrt{2}+\sqrt{3}}{2}$
$=notrational$
$b)\tan 22\frac{1}{2}+\cot 22\frac{1}{2}$
$=\tan 22\frac{1}{2}+\frac{1}{\tan 22\frac{1}{2}}$
$=\sqrt{2}-1+\frac{1}{\sqrt{2}-1}$
$=\frac{{(\sqrt{2}-1)}^2+1}{\sqrt{2}-1}=notrational$
$c)\sin 15-\cos 15$
$=\frac{\sqrt{3}-1}{2\sqrt{2}}-\frac{\sqrt{2}+\sqrt{3}}{2}$
$=notrational$
$d)\tan 22\frac{1}{2}-\cot 22\frac{1}{2}$
$=\tan 22\frac{1}{2}-\frac{1}{\tan 22\frac{1}{2}}$
$=\sqrt{2}-1-\frac{1}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}$
$=\sqrt{2}-1-\frac{\sqrt{2}+1}{2-1}$
$=\sqrt{2}-1-\sqrt{2}-1$
$=-2$
$rationalnumber$