Answer
The magnitude of the orbital angular momentum$(\vec L)$ of an electron trapped in an atom is given by
$L=\sqrt {l(l+1)}\hbar,\;\;\;\text{for}\;l=0,1,2,...,(n-1)$
The component $L_z$ of $\vec L$ on a $z$ axis is given by
$L_z=m_l\hbar,\;\;\;\text{for}\;m_l=0,±1,±2,...,±l$
If, $L_x$, $L_y$ are the components of $\vec L$ along $x$ and $y$ axes, then we can write
$L^2=L_x^2+L_y^2+L_z^2$
or, $L_x^2+L_y^2=L^2-L_z^2$
or, $L_x^2+L_y^2=(\sqrt {l(l+1)}\hbar)^2-(m_l\hbar)^2$
or, $L_x^2+L_y^2=[l(l+1)-m_l^2]\hbar^2$
or, $(L_x^2+L_y^2)^{1/2}=[l(l+1)-m_l^2]^{1/2}\hbar$ (proved)
Work Step by Step
The magnitude of the orbital angular momentum$(\vec L)$ of an electron trapped in an atom is given by
$L=\sqrt {l(l+1)}\hbar,\;\;\;\text{for}\;l=0,1,2,...,(n-1)$
The component $L_z$ of $\vec L$ on a $z$ axis is given by
$L_z=m_l\hbar,\;\;\;\text{for}\;m_l=0,±1,±2,...,±l$
If, $L_x$, $L_y$ are the components of $\vec L$ along $x$ and $y$ axes, then we can write
$L^2=L_x^2+L_y^2+L_z^2$
or, $L_x^2+L_y^2=L^2-L_z^2$
or, $L_x^2+L_y^2=(\sqrt {l(l+1)}\hbar)^2-(m_l\hbar)^2$
or, $L_x^2+L_y^2=[l(l+1)-m_l^2]\hbar^2$
or, $(L_x^2+L_y^2)^{1/2}=[l(l+1)-m_l^2]^{1/2}\hbar$ (proved)