Question

If α, β are different values of x satisfying the equation a cos x + b sinα x = c, where a, b and c are constants, then `tan ((alpha + beta)/2)` is

Options

• `a + b`

• `b - a`

• `a/b`

• `b/a`

Answer 1

`b/a`

Explanation:

Given `alpha, beta` are different values of x satisfying the equation a cos x + b sin x + c,

∴ a cos α + b sin α = c .......(i)

a cos β + b sin β = c ......(ii)

From (i) and (ii)

`a cos alpha + b sin alpha = alpha cos beta + b sin beta`

⇒ `a(cos alpha - cos beta) + b(sin alpha - sin beta)` = 0

⇒ `a(- 2 sin  (alpha + beta)/2 . sin  (alpha - beta)/2) + b(2 cos  (alpha + beta)/2 . sin  (alpha - beta)/2)` = 0

⇒ `2sin ((alpha - beta)/2) [b cos  (alpha + beta)/2 - a sin  (alpha + beta)/2]` = 0

⇒ `sin  (alpha - beta)/2 ≠ 0` .......[∴ α ≠ β]

∴ `b cos  (alpha + beta)/2 - a sin  (alpha + beta)/2` = 0

∴ `tan  (alpha + beta)/2 = b/a`