Trigonometric Functions — MATHS STD 11 Science — Question

2. $\text{x=y, x}\neq0$

Solution:

We have:

$\sec^2\text{x}=\frac{4\text{xy}}{(\text{x}+\text{y})^2}$

$\Rightarrow\frac{4\text{x}\text{y}}{(\text{x}+\text{y})^2}\geq1 $ $[\therefore\sec^2\text{x}\geq1]$

$\Rightarrow4\text{xy}\geq(\text{x}+\text{y})^2$

$\Rightarrow4\text{xy}\geq\text{x}^2+\text{y}^2+2\text{xy}$

$\Rightarrow2\text{xy}\geq\text{x}^2+\text{y}^2$

$\Rightarrow(\text{x}-\text{y})^2\leq0$

$\Rightarrow(\text{x}-\text{y})\leq0$

$\Rightarrow\text{x}=\text{y}$

For $\text{x}=0,\sec^2\text{x}$ will not be defined,

$\Rightarrow\text{x}\neq 0$

$\therefore\text{x}=\text{y}$