Expand the Trigonometric Expression 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 two fractions, which is sin(x)*cos(x)

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

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

(sin2(x)+cos2(x))/(sin(x)*cos(x))=1/(sin(x)*cos(x))

4. Use the double angle identity sin(2*x)=2*sin(x)*cos(x) which implies sin(x)*cos(x)=sin(2*x)/2 to further simplify the expression.

1/(sin(x)*cos(x))=1/sin(2*x)/2

5. Simplify the complex fraction and use the reciprocal identity 1/sin(2*x)=csc(2*x)

2/sin(2*x)=2*csc(2*x)

Final Answer

tan(x)+cot(x)=2*csc(2*x)

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