Pulling force making an angle $\theta $ to the horizontal is applied on a block of weight $W$ placed on a horizontal table. If the angle of friction is $\alpha $, then the magnitude of force required to move the body is equal to
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(c) $\uparrow: N+F \sin \theta=W \Rightarrow N=W-F \sin \theta$
The block will move if
$F \cos \theta \geq f_{\max }$
$F \cos \theta \geq \mu(W-F \sin \theta)$
$\mu=\tan \alpha=\frac{\sin \alpha}{\cos \alpha}$
$F \cos \theta \geq \frac{\sin \alpha}{\cos \alpha}(W-F \sin \theta)$
$F(\cos \theta \cos \alpha+\sin \theta \sin \alpha) \geq W \sin \alpha$
$F \geq \frac{W \sin \alpha}{\cos (\theta-\alpha)}$
$F_{\min }=\frac{W \sin \alpha}{\cos (\theta-\alpha)}$
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