Simplify tan(x)+cot(x)

Problem

tan(x)+cot(x)

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the definitions tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x)

tan(x)+cot(x)=sin(x)/cos(x)+cos(x)/sin(x)

2. Find a common denominator to combine the fractions, which is sin(x)*cos(x)

(sin(x)*sin(x))/(sin(x)*cos(x))+(cos(x)*cos(x))/(sin(x)*cos(x))

3. Combine the numerators over the common denominator.

(sin2(x)+cos2(x))/(sin(x)*cos(x))

4. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the numerator.

1/(sin(x)*cos(x))

5. Use reciprocal identities 1/sin(x)=csc(x) and 1/cos(x)=sec(x) to write the expression in its simplest form.

sec(x)*csc(x)

Final Answer

tan(x)+cot(x)=sec(x)*csc(x)

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