Question

Which of the following is true in a triangle ABC?

Options

• `(b + c)sin  (B - C)/2 = 2acos  A/2`

• `(b + c)cos  A/2 = 2asin  (B - C)/2`

• `(b - c)cos  A/2 = asin  (B - C)/2`

• `(b - c) sin  (B - C)/2 = 2acos  A/2`

Answer 1

`underline((b - c)cos  A/2 = asin  (B - C)/2)`

Explanation:

`(b - c)/a = (sin B - sinC)/sinA`

= `(2sin  (B - C)/2 cos  (B + C)/2)/(2sin  A/2 cos  A/2)`

=  `(sin((B - C)/2)cos((B + C)/2))/(cos((B + C)/2)cos  A/2`

= `(sin  (B - C)/2)/(cos  A/2)`

⇒ `(b - c)cos  A/2 = a sin  (B - C)/2`