If cosx/(1+sinx) = |tanx|, then find the number of solutions in [0, 2π].

Solution:

Answer: x = π/6

Case 1: tan x ≥ 0

cosx/(1+sinx ) = sinx/cosx

⇒cos2x = sin2x+sinx

⇒1-sin2x = sin2x + sinx

⇒2 sin2x + sin x – 1 = 0

⇒sin x = -1 ,1/2

sin x = 1/2 ⇒ x = π/6

sin x = -1 (not possible)

Case 2: tan x < 0

(Cos x)/(1+sinx) = -sin x/cos x

⇒cos2x = -sin2x – sinx

⇒1-sin2x = -sin2x – sinx

Sin x = -1 (Not possible)

Only one solution exists x = π/6

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