Question

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

Answer 1

Range of $|\sin x|+|\cos x|$ is $[1,\sqrt{2}]$
So, $y=[|\sin x|+|\cos x|]=1$
$\Rightarrow$ slope of tangent at any point is $0.$(due to constant function)
$\Rightarrow m_1=0$
$x^2+y^2=5$
on differentiating w.r.t. $x$, we get
$\frac{dy}{dx}=-\frac{x}{y}=\text{slope of tangent}$
at ($y=1$ $\Rightarrow x=\pm 2)$
$\Rightarrow \frac{dy}{dx}=\pm 2=m_2$
Let $\theta$ be the angle between the curves
$\therefore \tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|=|m_2|$
$\Rightarrow {\text{cosec}}^2\theta =\frac{5}{4}$