Question

If A, B and C are the angles of a triangle, such that sec (A – B), sec A and sec (A + B) are in arithmetic progression, then ______.

Options

• cosec² A = `2  "cosec"^2  B/2`

• 2 sec² A = `sec^2  B/2`

• 2 cosec² A = `"cosec"^2  B/2`

• 2 sec² B = `sec^2  A/2`

Answer 1

If A, B and C are the angles of a triangle, such that sec (A – B), sec A and sec (A + B) are in arithmetic progression, then `underlinebb(2 sec^2 A = sec^2  B/2)`.

Explanation:

∵ sec A = `(sec(A - B) + sec(A + B))/2`

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

= `(cos A cos B)/([cos^2A cos^2B - sin^2A sin^2B])`

`\implies` sec A = `(cos A cos B)/([cos^2 A cos^2 B - (1 - cos^2 A)(1 - cos^2 B)])`

`\implies` cos² A + cos² B – 1 = cos² A cos B

`\implies` cos² A = 1 + cos B

`\implies` cos² A = `2 cos^2  B/2`

`\implies` sec² A = `1/2 sec^2  B/2`

`\implies` 2 sec² A = `sec^2  B/2`