Question

For any real $\theta$ the maximum value of ${\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)$ is

• A. $1$
• B. $1+{\sin }^{2}1$
• C. $1+{\cos }^{2}1$
• D. $2+{\cos }^{2}1$

Answer 1

The correct option is B $1+{\sin }^{2}1$

$f\left(\theta \right)=co{s}^2\left(cos\theta \right)+si{n}^2\left(sin\theta \right)$
$-1\le cos\theta \le 1$
$-1\le sin\theta \le 1$
$\therefore sin\theta$ in (-1,1) is increasing function
So, $sin1>sin0$
$\therefore cos\theta$ in (0,1) is decreasing function
$cos1<cos0$
$\therefore$ for maximum $f\left(\theta \right)$
$sin\theta =1andcos\theta =0$
$\therefore f\left(\theta \right)=(cos\theta {)}^2+si{n}^2(1)$
$=1+si{n}^{2}1$