Question

State whether the following statements are true or false.
The inequality $2^{{\sin }^2\theta }+2^{{\cos }^2\theta }\ge 2\sqrt{2}$ holds for all real $\theta$.

• A. True
• B. False

Answer 1

The correct option is A True

$2^{{\sin }^2\theta }+2^{{\cos }^2\theta }\ge 2\sqrt{2}$

we know that ,

$AM\ge GM$

$\Rightarrow \frac{2^{{\sin }^2\theta }+2^{{\cos }^2\theta }}{2}\ge \sqrt{2^{{\sin }^2\theta }{.2}^{{\cos }^2\theta }}$

$\Rightarrow \frac{2^{{\sin }^2\theta }+2^{{\cos }^2\theta }}{2}\ge \sqrt{2^{{\sin }^2\theta +{\cos }^2\theta }}$

$\Rightarrow \frac{2^{{\sin }^2\theta }+2^{{\cos }^2\theta }}{2}\ge \sqrt{2}$

$\Rightarrow 2^{{\sin }^2\theta }+2^{{\cos }^2\theta }\ge 2\sqrt{2}$

Hence given inequality is true.