In a triangle ABC, sec A (sin B cos C + cos B sin C) equals:

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AAI ATC Junior Executive 27 July 2022 Shift 3 Official Paper
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1. c/a
2. 1
3. tan A
4. cot A

Answer (Detailed Solution Below)

Option 3 : tan A
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Detailed Solution

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, sum of 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).

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