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

Problem

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

Solution

1. Rewrite the terms on the left side using the fundamental trigonometric identities tan(x)=sin(x)/cos(x) and 1/tan(x)=cot(x)=cos(x)/sin(x)

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

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

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

3. Combine the fractions over the common denominator.

(cos2(x)+sin2(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

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

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