Verify the Identity tan(x)+cot(x)=1/(sin(x)cos(x))

Problem

tan(x)+cot(x)=1/(sin(x)*cos(x))

Solution

1. Express the left side of the identity in terms of sine and cosine using the quotient identities 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 for the two fractions, which is sin(x)*cos(x)

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

3. Combine the fractions 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))

Final Answer

tan(x)+cot(x)=1/(sin(x)*cos(x))

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