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Question:

Let S = {x ∈ (−π, π) : x ≠ 0, ±π/2}. The sum of all distinct solutions of the equation √3secx + cosecx + 2(tanx − cotx) = 0 in the set S is equal to

π/9

5π/9

−π/9

0

Solution:

√3secx+cosecx+2(tanx−cotx)=0⇒√3sinx+cosx+2(sin2x−cos2x)=0√3sinx+cosx𕒶cos2x=0⇒sin(x+π3)=cos2xcos(π3−x)=cos2x⇒2x=2nπ±(π3−x)x=2nπ3+π9orx=2nπ−π3x=𕒵00o,𕒺00,200,1400s0𕒵00o𕒺0o+20o+140o=0