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√3 Cos x - Sin x = 1

divide the equation by 2:

=> √3/2 Cos x - 1/2 sin x = 1/2

=> Sin π/3 Cos x - Cos π/3 Sin x = Sin π/6

=>**Sin (π/3 - x) = Sin π/6**

=>*Sin (x - π/3) = Sin (-π/6) or Sin (π + π/6)*

=> x - π/3 = 2 n π - π/6 or 2 n π + 7 π/6

* x = 2 n π + π/6 or 2 n π + 3 π / 2*

this 2 n π + 3 π/2 can also be expressed as: 2 n π + 2π - π/2

ie., 2 m π - π / 2

*Hence, x = 2 n π + π/6 and x = 2 n π - π/2*

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another way:

√3 Cos x - Sin x = 1

=> √3/2 Cos x - 1/2 Sin x = 1/2

=> Cos π/6 Cos x - Sin π/6 Sin x = Cos π/3

=>*Cos (x + π/6) = Cos π/3*

* we have general solution for Cos X = Cos Y, X = 2 n π + Y or 2 n π - Y*

=> x + π/6 = 2 n π + π/3 or 2 n π - π/3

*=> x = 2 n π + π/6 or 2 n π - π/2*

divide the equation by 2:

=> √3/2 Cos x - 1/2 sin x = 1/2

=> Sin π/3 Cos x - Cos π/3 Sin x = Sin π/6

=>

=>

=> x - π/3 = 2 n π - π/6 or 2 n π + 7 π/6

ie., 2 m π - π / 2

==========================

another way:

√3 Cos x - Sin x = 1

=> √3/2 Cos x - 1/2 Sin x = 1/2

=> Cos π/6 Cos x - Sin π/6 Sin x = Cos π/3

=>

=> x + π/6 = 2 n π + π/3 or 2 n π - π/3