Question

If ${\sin }^2\theta {\cos }^2\theta =x$, then

• A. $x<\frac{1}{4}$
• B. $x\le \frac{1}{4}$
• C. $x>\frac{1}{4}$
• D. $x\ge \frac{1}{4}$

Answer 1

The correct option is B $x\le \frac{1}{4}$

${\sin }^2\theta {\cos }^2\theta ={\sin }^2\theta (1-{\sin }^2\theta )$

$={\sin }^2\theta -{\sin }^4\theta$

$=\frac{1}{4}-\frac{1}{4}+{\sin }^2\theta -{\sin }^4\theta$

$=\frac{1}{4}-{(\frac{1}{2}-{\sin }^2\theta )}^2\le \frac{1}{4}$

Since ${(\frac{1}{2}-{\sin }^2\theta )}^2\ge 0)$