Problem

Problem

Find the number of solutions of the equations; (i) ∣cot⁡x∣=cot⁡x+1sin⁡x|\cot x| = \cot x + \frac{1}{\sin x} , when x∈[0,2π]x \in [0, 2\pi] (ii) sin⁡3xcos⁡x+sin⁡2x⋅cos⁡2x+sin⁡x⋅cos⁡3x=1\sin^3 x \cos x + \sin^2 x \cdot \cos^2 x + \sin x \cdot \cos^3 x = 1 , when x∈[0,2π]x \in [0, 2\pi] (iii) 2cos⁡x=∣sin⁡x∣2^{\cos x} = |\sin x| , when x∈[−2π,2π]x \in [-2\pi, 2\pi] (iv) ∣cos⁡x∣=[x]|\cos x| = [x] , (where [.][. ] denotes the greatest integer function). (v) x+2tan⁡x=π2x + 2 \tan x = \frac{\pi}{2} , when x∈[0,2π]x \in [0, 2\pi] .

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