Concept:

• sin (A + B) = sin A cos B + cos A sin B ___(1)
• The sum of the three angles of a triangle is 180°
   
• sin(180 - θ) = sin θ  ___(2)

Calculation:

In triangle ABC, the sum of the angles = A + B + C = 180° 

⇒ B + C = 180 - A   ___(3)

Given,  sec A (sin B cos C + cos B sin C)

⇒ sec A sin (B + C)     ∵ Using {(1)}

⇒ sec A. sin (180 - A)   ∵ Using {(2)}

⇒ sec A. sin A   ∵ Using {(3)}

⇒  \({\sin A \over \cos A }\)

⇒ tan A

∴ The correct answer is option (3).