Question

If $P=\frac{\sin {300}^{\circ }\cdot \tan {330}^{\circ }\cdot \sec {420}^{\circ }}{\tan {135}^{\circ }\cdot \sin {210}^{\circ }\cdot \sec {315}^{\circ }}$ and $Q=\frac{\sec {480}^{\circ }\cdot \text{cosec}{570}^{\circ }\cdot \tan {330}^{\circ }}{\sin {600}^{\circ }\cdot \cos {660}^{\circ }\cdot \cot {405}^{\circ }},$ then the value of $P$ and $Q$ are respectively

• A. $\sqrt{2},-\frac{16}{3}$
• B. $\sqrt{2},\frac{16}{3}$
• C. $-2,\frac{3}{16}$
• D. $\sqrt{2},\frac{3}{16}$

Answer 1

The correct option is B $\sqrt{2},\frac{16}{3}$

$P=\frac{\sin {300}^{\circ }\cdot \tan {330}^{\circ }\cdot \sec {420}^{\circ }}{\tan {135}^{\circ }\cdot \sin {210}^{\circ }\cdot \sec {315}^{\circ }}$
$\Rightarrow P=\frac{\sin ({360}^{\circ }-{60}^{\circ })\cdot \tan ({360}^{\circ }-{30}^{\circ })\cdot \sec ({360}^{\circ }+{60}^{\circ })}{\tan ({90}^{\circ }+{45}^{\circ })\cdot \sin ({180}^{\circ }+{30}^{\circ })\cdot \sec ({360}^{\circ }-{45}^{\circ })}$
$\Rightarrow P=\frac{-\sin {60}^{\circ }\cdot (-\tan {30}^{\circ })\cdot \left(\sec {60}^{\circ }\right)}{-\cot {45}^{\circ }\cdot (-\sin {30}^{\circ })\cdot \left(\sec {45}^{\circ }\right)}\Rightarrow P=\frac{\left(\frac{-\sqrt{3}}{2}\right)\cdot \left(\frac{-1}{\sqrt{3}}\right)\cdot \left(2\right)}{(-1)\cdot \left(\frac{-1}{2}\right)\cdot \left(\sqrt{2}\right)}\Rightarrow P=\sqrt{2}$

$Q=\frac{\sec {480}^{\circ }\cdot \text{cosec}{570}^{\circ }\cdot \tan {330}^{\circ }}{\sin {600}^{\circ }\cdot \cos {660}^{\circ }\cdot \cot {405}^{\circ }}$
$\Rightarrow Q=\frac{\sec ({540}^{\circ }-{60}^{\circ })\cdot \text{cosec}({540}^{\circ }+{30}^{\circ })\cdot (\tan {360}^{\circ }-{30}^{\circ })}{\sin ({540}^{\circ }+{60}^{\circ })\cdot \cos ({720}^{\circ }-{60}^{\circ })\cdot \cot ({360}^{\circ }+{45}^{\circ })}$
$\Rightarrow Q=\frac{(-\sec {60}^{\circ })\cdot (-\text{cosec}{30}^{\circ })\cdot (-\tan {30}^{\circ })}{(-\sin {60}^{\circ })\cdot \left(\cos {60}^{\circ }\right)\cdot \left(\cot {45}^{\circ }\right)}$
$\Rightarrow Q=\frac{(-2)(-2)\left(\frac{-1}{\sqrt{3}}\right)}{\left(\frac{-\sqrt{3}}{2}\right)\cdot \left(\frac{1}{2}\right)\cdot \left(1\right)}\Rightarrow Q=\frac{16}{3}$

Hence, $P=\sqrt{2},Q=\frac{16}{3}$