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When I study the domain my function f:

f[x_] := 2/(Cos[x]^2) + 1/Log[Sin[x]]
FunctionDomain[f[x], x, Reals]

I receive the following answer.

1/2 + x/π ∉ Integers && 
C[1] ∈ Integers && (2 π C[1] < x < 1/2 (π + 4 π C[1]) || 
1/2 (π + 4 π C[1]) < x < π + 2 π C[1])

I find it difficult to interpret the beginning of the answer:

1/2 + x/π ∉ Integers 

Please explain this part. For the rest, I'm ok.

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  • $\begingroup$ @Bendesarts If x had the value of [Pi]/2 the expression 1/2 + x/[Pi] would evaluate to 1, an integer. The expression says that it is prohibited for 1/2 + x/[Pi] to evaluate to be an integer or you will be outside the function domain. $\endgroup$ – Jack LaVigne Nov 7 '15 at 16:04
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Since your f[x] is a sum of two functions, why not try them separately?

FunctionDomain[2/(Cos[x]^2), x, Reals]

1/2 + x/π ∉ Integers

This is saying that values of x for which the denominator is zero are forbidden. The other half is

FunctionDomain[1/Log[Sin[x]], x, Reals]

which gives the rest.

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