Question

The maximum value of the expression, $|\sqrt{{\sin }^{2}x+2a^2-}\sqrt{2a^2-1-{\cos }^{2}x}|$ where a and x are real numbers is :

• A. $\sqrt{3}$
• B. $\sqrt{2}$
• C. $1$
• D. none of these

Answer 1

Answer: (B)

$\sqrt{2}$

The expression $|\sqrt{{\sin }^{2}x+2a^2-}\sqrt{2a^2-1-{\cos }^{2}x|}$ can also be written as $|\sqrt{{\sin }^{2}x+2a^2-}\sqrt{2a^2+{\sin }^{2}x-2}$
Let us take ${\sin }^{2}x+2a^2$ to be y.
Thus, the expression looks like $\sqrt{y}-\sqrt{y-2}$
Since both the terms are positive, the maximum value will be when y = 2 since that makes the second term to be 0.
Thus, maximum value is $\sqrt{2}$