Correct option is: (C) \(90^\circ\)
\(\tan 35^{\circ} \times \tan 55^{\circ}=\sin A \)
\(\Rightarrow \frac{1}{\cot 35^{\circ}} \times \tan 55^{\circ}=\sin A \quad \left[\because \tan \theta=\frac{1}{\cot \theta}\right] \)
\(\Rightarrow \frac{1}{\cot \left(90^{\circ}-55^{\circ}\right)} \times \tan 55^{\circ}=\sin A \)
\(\Rightarrow \frac{1}{\tan 55^\circ}\times \tan 55^{\circ}=\sin A \quad\left[\tan \theta=\cot \left(90^{\circ}-\theta\right)\right] \)
\(1=\sin A \)
\( \sin A=\sin 90^{\circ} \)
\(\therefore A=90^{\circ}
\)
