Question

For the given two statements, select the correct option.
Statement $\left(1\right):$
Maximum value of ${\sin }^{2}x+{\cos }^{2}y\text{is}2$

Statement $\left(2\right):$
Maximum value of $2{\sec }^{2}z\text{is}2$

Answer 1

Statement $(1:)$ Maximum value of ${\sin }^{2}x+{\cos }^{2}y$ is $2$
Now, the maximum value of the expression occurs when ${\sin }^{2}x\&{\cos }^{2}y$ simultaneously attains its maximum value as both are independent terms.

Now, the maximum value of ${\sin }^{2}x\text{and}{\cos }^{2}y\text{are}1.$

$\Rightarrow \text{Maximum value of}{\sin }^{2}x+{\cos }^{2}y=1+1=2$

$\therefore$ Statement $1$ is correct.

Now consider ${\sec }^{2}z$
$\sec z\in (-\infty ,-1]\cup [1,\infty )$
$\Rightarrow {\sec }^{2}z\in [1,\infty )$
$\Rightarrow 2{\sec }^{2}z\in [2,\infty )$
$\therefore$ Statement $2$ is wrong.