Question

For any real $\theta$, 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. non of these

Answer 1

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

${\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)$ is maximum when $\theta ={90}^o$ as cos is a decreasing function and sin is a increasing function

So, ${\cos }^2\left(\cos {90}^o\right)+{\sin }^2\left(\sin {90}^o\right)={\cos }^2\left(0\right)+{\sin }^{2}1=1+{\sin }^{2}1$

Therefore, Answer is $1+{\sin }^{2}1$