Question

If $0<\theta <\pi$ then the minimum value of ${\sin }^5\theta +{cosec}^5\theta$ is

• A. $0$
• B. $1$
• C. $2$
• D. None of the above.

Answer 1

The correct option is C $2$

we know that, $A.M.\ge G.M.$

$⟹\frac{a+b}{2}\ge \sqrt{ab}$

let $a=si{n}^5\theta$ and $b=cose{c}^5\theta$

therefore $\frac{si{n}^5\theta +cose{c}^5\theta }{2}\ge \sqrt{si{n}^5\theta \ast cose{c}^5\theta }$

$⟹si{n}^5\theta +cose{c}^5\theta \ge 2\sqrt{(sin\theta \ast cosec\theta {)}^5}$

$⟹si{n}^5\theta +cose{c}^5\theta \ge 2\sqrt{(sin\theta \ast cosec\theta {)}^5}$

$⟹si{n}^5\theta +cose{c}^5\theta \ge 2\sqrt{(sin\theta \ast (\frac{1}{sin\theta }){)}^5}$

$⟹si{n}^5\theta +cose{c}^5\theta \ge 2\sqrt{1^5}$

$⟹si{n}^5\theta +cose{c}^5\theta \ge 2$

therefore the minimum value of $si{n}^5\theta +cose{c}^5\theta$ is $2$