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# Determining the define range?

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I need to determine the defined range for $${1+tanθ \over 1 + cotθ}$$=$${1-tanθ \over cotθ-1}$$, how should I do so?

The answer is θ (cannot equal to) kπ/4, k E Z

Jun 11, 2020

### 2+0 Answers

#1
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actually any value of $$θ$$ is true if $$θ$$ is a variable but the equation $${1+\tanθ \over 1 + \cotθ}$$=$${1-\tanθ \over \cotθ-1}$$ is an identity

Jun 11, 2020
#2
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You need to eliminate all the possibilities for a denominator to be zero.

Since there is a  tan(x)  term, (and since  tan(x) = sin(x)/cos(x) ), you must eliminate all the values that

make  cos(x) = 0.

Since there is a  cot(x)  term, (and since  cot(x) = cos(x)/sin(x) ), you must eliminate all the values that

make  sin(x) = 0.

Since there is a denominator of  1 + cot(x),  you must eliminate all the values that make  cot(x) = -1.

Since there is a denominator of  cot(x) - 1,  you must eliminate all the values that make  cot(x) = 1.

[And, remember that both  tan(x)  and  cot(x)  have a period of  pi.]

Jun 11, 2020