Question

Let $S={\sin }^2{30}^{\circ }+{\sin }^2{45}^{\circ }+{\sin }^2{60}^{\circ }$ and $P=co{\sec }^2{45}^{\circ }.{\sec }^2{30}^{\circ }.{\sec }^3{0}^{\circ }.\cos {60}^{\circ }$. Then the correct statement is:

• A. $S<P$
• B. $S=P$
• C. $SP=2$
• D. $S+P>3$

Answer 1

Answer: (C)

$SP=2$

$S=\frac{1}{4}+\frac{1}{2}+\frac{3}{4}=\frac{3}{2}$

$P=2\times \frac{4}{3}\times 1\times \frac{1}{2}=\frac{4}{3}$

$\Rightarrow S>P$

Now, $SP=\frac{3}{2}\times \frac{4}{3}=2$

$S+P=\frac{3}{2}+\frac{4}{3}=\frac{9+8}{6}=\frac{17}{6}<3$

Hence, option C is correct.