The minimum value of α for which the equation 4 sin x + 1 1 - sin x = α has at least one solution in 0 , π 2 is______.

Step-by-step solution

Let f x = 4 sin x + 1 1 - sin x ⇒ f x = 4 sin x + 1 + sin x 1 - sin 2 x ⇒ f x = 4 cosec x + sec 2 x + tan x sec x ⇒ f ' x = - 4 cos x sin 2 x + 2 sin x + 1 cos 3 x + sin 2 x cos 3 x ⇒ f ' x = - 4 cos x sin 2 x + sin x + 1 2 cos 3 x ⇒ f ' ' x = - 4 - sin 3 x - 2 sin x cos 2 x sin 4 x + 2 sin x + 1 cos 4 x + 3 cos 2 x sin x sin x + 1 2 cos 6 x ⇒ f ' ' x = - 4 - sin 3 x - 2 sin x 1 - sin 2 x sin 4 x + 2 sin x + 1 1 - sin 2 x 2 + 3 1 - sin 2 x sin x sin x + 1 2 1 - sin 2 x 3 For