\(dm = k \sec ^{2} \theta d \theta\)

\(\Rightarrow \frac{ dm }{ m }=\frac{ k \sec ^{2} \theta}{ k \tan \theta} d \theta\)

\(\Rightarrow \frac{ dm }{ m }=\frac{ d \theta}{\sin \theta \cos \theta}=\frac{2 d \theta}{\sin 2 \theta}\)

\(\Rightarrow \%\) error is minimum when \(\sin 2 \theta\) has maximum value

\(2 \theta=\frac{\pi}{2}\) or \(\theta=45^{\circ}\)