(3) \( \sin A+\tan \left(360^{\circ}-A\right) \cdot \sin \left(90^{\circ}-A\right) \)

Let us simplify the expression step by step. 1. Rewrite the expression: \[ \sin A + \tan\left(360^\circ - A\right) \cdot \sin\left(90^\circ - A\right) \] 2. Simplify \(\tan\left(360^\circ-A\right)\): Since the tangent function has a period of \(180^\circ\), we have \[ \tan\left(360^\circ-A\right) = \tan\left(-A\right) \] and using the fact that tangent is an odd function, \[ \tan\left(-A\right) = -\tan A. \] 3. Simplify \(\sin\left(90^\circ-A\right)\): Using the co-function identity, \[ \sin\left(90^\circ-A\right) = \cos A. \] 4. Substitute these results into the expression: \[ \sin A + \left(-\tan A\right) \cdot \cos A. \] 5. Write the expression as: \[ \sin A - \tan A \cdot \cos A. \] 6. Express \(\tan A\) in terms of \(\sin A\) and \(\cos A\): \[ \tan A = \frac{\sin A}{\cos A}. \] Substitute into the expression: \[ \sin A - \frac{\sin A}{\cos A} \cdot \cos A. \] 7. Simplify the product: \[ \frac{\sin A}{\cos A} \cdot \cos A = \sin A. \] Thus, the expression becomes: \[ \sin A - \sin A. \] 8. Finally, subtract: \[ \sin A - \sin A = 0. \] The simplified result is: \[ 0 \]