Question

If $\theta$ is angle between curves $y=[|\sin x|+|\cos x|]$, $\left(\right[\cdot \left]\text{denotes greatest integer function}\right)$ and $x^2+y^2=5$ then $4{\text{cosec}}^2\theta$ is​​​​​​​

Answer 1

Range of $|\sin x|+|\cos x|$ is $[1,\sqrt{2}]$
So, $y=[|\sin x|+|\cos x|]=1$
$x^2+y^2=5$
on differentiating w.r.t $x$, we get
$\frac{dy}{dx}=-\frac{x}{y}$ (slope of tangent)
at $y=1$ $\Rightarrow x=\pm 2$
$\Rightarrow (\frac{dy}{dx}=\pm 2)$
$\Rightarrow \tan \theta =\pm 2$
$\Rightarrow {\text{cosec}}^2\theta =\frac{5}{4}$