Number of values of $ x \in(-\pi, 3 \pi)-\left\{\frac{-\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}, 2 \pi, \frac{5 \pi}{2}\right\} $ satisfying the equation $ \tan x \cot \frac{x}{2}=2-\cot x \tan \frac{x}{2} $ is/are

Question

Number of values of $ x \in(-\pi, 3 \pi)-\left\{\frac{-\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}, 2 \pi, \frac{5 \pi}{2}\right\} $ satisfying the equation $ \tan x \cot \frac{x}{2}=2-\cot x \tan \frac{x}{2} $ is/are,

MARKS App · Accepted Answer

tan x cot x 2 + tan x 2 cot x = 2 Let tan x cot x 2 = t ⇒ t + 1 t = 2 ⇒ t - 1 2 = 0 ⇒ t = 1 ⇒ tan x cot x x = 1 ⇒ sin x cos x 2 = cos x sin x 2 ⇒ sin x 2 = 0 ⇒ x 2 = 0 , π as x 2 ∈ - π 2 , 3 π 2 ⇒ x = 0 , 2 π both are not possible.

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