Question

Which of the following statements 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. $\cos \frac{7\theta }{8}\ge (\cos \theta {)}^{7/8}$

Answer 1

Answer: (A), (C)

The correct options are
A $(\cos \theta {)}^{1/2}\le \cos \frac{\theta }{2}$
C $\cos \frac{5\theta }{6}\ge (\cos \theta {)}^{5/6}$

$f\left(\theta \right)={\cos }^n\theta \&g\left(\theta \right)=\cos n\theta ,0<\theta <\frac{\pi }{2}$

As $0<n<1,0<n\theta <\frac{\pi }{2}.$ So, $\cos n\theta$ ranges from $1$ to 0 whereas ${\cos }^n\theta$ ranges from $1$ to $0.$

Also, $f^{\prime }\left(\theta \right)<g^{\prime }\left(\theta \right).$

$\therefore {\cos }^{1/2}\theta <\cos \frac{\theta }{2}\&{\cos }^{5/6}\theta \le \cos \frac{5\theta }{6}$

Hence, the answers are ${\cos }^{1/2}\theta <\cos \frac{\theta }{2}\&{\cos }^{5/6}\theta \le \cos \frac{5\theta }{6}.$