Question

Which of the following statement is/are correct for $0<\theta <\frac{\pi }{2}$?

• A. $(\cos \theta {)}^{1/2}\le \cos \frac{\theta }{2}$
• B. $(\cos \theta {)}^{3/4}\ge \cos \frac{3\theta }{4}$
• C. $\cos \frac{5\theta }{6}\ge (\cos \theta {)}^{5/6}$
• D. $\cos \frac{7\theta }{8}\le (\cos \theta {)}^{7/8}$

Answer 1

Answer: (A), (C)

$(\cos \theta {)}^{1/2}\le \cos \frac{\theta }{2}$
$\cos \frac{5\theta }{6}\ge (\cos \theta {)}^{5/6}$

$0<\theta <\frac{\pi }{2}$

$\Rightarrow 0<\frac{\theta }{n}<\frac{\pi }{2n}$

In the range $(0,\frac{\pi }{2}),$ $cos\left(\frac{\theta }{n}\right)$ ranges from $1$ to $\text{some positive value}$ whereas $cos\theta$ ranges from $1$ to $0$

So, $cos\left(\frac{\theta }{n}\right)>cos\left(\theta \right)\Rightarrow cos\left(\frac{\theta }{n}\right)>(cos\theta {)}^{1/n}$

So, Options (A),(C)