What is the domain of (cos(x))/(tan(x)-1) ?

The domain of (cos(x))/(tan(x)-1) is

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domain of (cos(x))/(tan(x)-1)

domain

\frac{cos ( x )}{tan ( x ) - 1}

2 πn \leq x < \frac{π}{4} + 2 πn or \frac{π}{4} + 2 πn < x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{5 π}{4} + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Interval Notation

[2 πn, \frac{π}{4} + 2 πn) \cup (\frac{π}{4} + 2 πn, \frac{π}{2} + 2 πn) \cup (\frac{π}{2} + 2 πn, \frac{5 π}{4} + 2 πn) \cup (\frac{5 π}{4} + 2 πn, \frac{3 π}{2} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, 2 π + 2 πn)

Solution steps

Take the denominator(s) of

\frac{cos ( x )}{tan ( x ) - 1}

and compare to zero:

x = \frac{π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Periodicity of

\frac{cos ( x )}{tan ( x ) - 1} : 2 π

Combine period:

2 πn \leq x < 2 π + 2 πn

and undefined points.

The domain is

2 πn \leq x < \frac{π}{4} + 2 πn or \frac{π}{4} + 2 πn < x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{5 π}{4} + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

What is the domain of (cos(x))/(tan(x)-1) ?
The domain of (cos(x))/(tan(x)-1) is