Determine value of real parameter $\lambda$ such that equation $$\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda $$has root in interval $\left(0,\frac{\pi}{2}\right)$
Hint:
\[\sin x+\cos x=\lambda \sin x\cos x,\sin 2x=t\Rightarrow {{\lambda }^{2}}{{t}^{2}}-4t-4=0\]We must have real solutions, and $t\in \left( 0,1 \right)$
We can see that $\sin x,\cos x>0$.
By Power Mean, we have
$$\frac2{\frac1{\sin x}+\frac1{\cos x}}\le\sqrt{\frac{\sin^2x+\cos^2x}2}\implies\frac1{\sin x}+\frac1{\cos x}\ge2\sqrt2$$and equality occurs when $x=\frac\pi4$. This means, we must have $\lambda\ge2\sqrt2$.
Furthermore, we can see that $\frac1{\sin x}+\frac1{\cos x}$ is a continuous function, and that $\lim_{x\to0^+}\frac1{\sin x}+\frac1{\cos x}=+\infty$.
Therefore, by IVT, there always exist $x\in\left(0,\frac{\pi}{4}\right)$ such that $\frac1{\sin x}+\frac1{\cos x}=\lambda$
The answer is, there will be a root if and only if $\boxed{\lambda\ge2\sqrt2}$.