Question

The value of $2{\sin }^2\frac{\pi }{6}+{\text{cosec}}^2\frac{7\pi }{6}\cdot {\cos }^2\frac{\pi }{3}$ is

• A. $\frac{1}{2}$
• B. $-\frac{3}{2}$
• C. $\frac{3}{2}$
• D. $1$

Answer 1

The correct option is C $\frac{3}{2}$

We know that,
$\text{cosec}\frac{7\pi }{6}=\text{cosec}(\pi +\frac{\pi }{6})=-\text{cosec}\frac{\pi }{6}$

Now,
$2{\sin }^2\frac{\pi }{6}+{\text{cosec}}^2\frac{7\pi }{6}\cdot {\cos }^2\frac{\pi }{3}$
$=2{\left(\frac{1}{2}\right)}^2+{(-\text{cosec}\frac{\pi }{6})}^2\cdot {\cos }^2\frac{\pi }{3}$
$=2\times \frac{1}{4}+2^2\times {\left(\frac{1}{2}\right)}^2$
$=\frac{1}{2}+1=\frac{3}{2}$