$\begin{array}{l}
we\,get\\
F\cos  = \mu \left( {mg - F\sin \theta } \right)\\
 = \mu \,mg - \,\mu F\sin \theta \\
or,\,F\,\left( {\cos \theta  + \mu \sin \theta } \right) = \mu mg\\
or,\,\,F = \frac{{\mu mg}}{{\cos \theta  + \mu \sin \theta }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {iii} \right)\\
For\,F\,to\,be\,m{\rm{inimum,}}\,{\rm{the}}\,{\rm{denominator}}\\
\left( {\cos \theta  + \mu \sin \theta } \right)\,should\,be\,{\rm{maximum}}.
\end{array}$

$\begin{array}{l}
\therefore \,\frac{d}{{d\theta }}\left( {\cos \theta  + \mu \sin \theta } \right) = 0\\
or,\, - \sin \theta  + \mu \cos \theta  = 0\\
or,\,\tan \theta  = \mu \\
or,\theta  = {\tan ^{ - 1}}\left( \mu  \right)\\
Then,\,\sin \theta \, = \frac{\mu }{{\sqrt {1 + {\mu ^2}} }}\,and
\end{array}$

$\begin{array}{l}
\cos \theta  = \frac{1}{{\sqrt {1 + {\mu ^2}} }}\\
Hence,\,{F_{\min }}\\
 = \frac{{\mu w}}{{\frac{1}{{\sqrt {1 + {\mu ^2}} }} + \frac{{{\mu ^2}}}{{\sqrt {1 + {\mu ^2}} }}}} = \frac{{\mu w}}{{\sqrt {1 + {\mu ^2}} }}
\end{array}$