Pls show the equality f(x+2pie)=f(x) if f(x)=1/(3+cos x)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to test if the function ` f(x + 2pi)` is equal to the function f(x), hence, you need to replace `x + 2pi` for `x` in equation of `f(x)` and then, you need to compare the equation resulted with the equation of `f(x)` , such that:

`f(x + 2pi) = 1/(3 + cos(x + 2pi))`

Expanding `cos(x + 2pi)` yields:

`cos(x + 2pi) = cos x*cos 2pi - sin x*sin 2pi`

Since `cos 2pi = 1` and `sin 2pi = 0` , yields:

`cos(x + 2pi) = cos x`

Replacing `cos x` for `cos(x + 2pi)` yields:

`f(x + 2pi) = 1/(3 + cos x) = f(x)`

Hence, comparing the equations of `f(x + 2pi)` and `f(x)` yields that that `f(x + 2pi) = f(x) = 1/(3 + cos x)` and the functions coincide.

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