Simplify (tan(x)+cot(x))/(cot(x))

Problem

(tan(x)+cot(x))/cot(x)

Solution

1. Split the fraction by dividing each term in the numerator by the denominator.

tan(x)/cot(x)+cot(x)/cot(x)

2. Simplify the second term because any non-zero expression divided by itself equals 1.

tan(x)/cot(x)+1

3. Use the reciprocal identity for the cotangent function, where cot(x)=1/tan(x)

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

4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

tan(x)⋅tan(x)+1

5. Apply the exponent rule to combine the tangent terms.

tan2(x)+1

6. Apply the Pythagorean identity where 1+tan2(x)=sec2(x)

sec2(x)

Final Answer

(tan(x)+cot(x))/cot(x)=sec2(x)

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