Verify the Identity

Problem

(tan(x)−1)/(tan(x)+1)=(1−cot(x))/(1+cot(x))

Solution

1. Identify the relationship between the trigonometric functions tan(x) and cot(x)

cot(x)=1/tan(x)

2. Substitute the identity for cot(x) into the right side of the equation.

(1−1/tan(x))/(1+1/tan(x))

3. Find a common denominator for the numerator and the denominator of the complex fraction.

(tan(x)/tan(x)−1/tan(x))/(tan(x)/tan(x)+1/tan(x))

4. Combine the terms in the numerator and the denominator.

(tan(x)−1)/tan(x)/(tan(x)+1)/tan(x)

5. Simplify the complex fraction by multiplying by the reciprocal or canceling the common denominator tan(x)

(tan(x)−1)/(tan(x)+1)

6. Conclude that the left side and the right side are identical.

(tan(x)−1)/(tan(x)+1)=(tan(x)−1)/(tan(x)+1)

Final Answer

(tan(x)−1)/(tan(x)+1)=(1−cot(x))/(1+cot(x))

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