Verify the Identity 1/(tan(x))+tan(x)=sec(x)csc(x)

Problem

1/tan(x)+tan(x)=sec(x)*csc(x)

Solution

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

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

3. Combine the fractions into a single expression.

(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))

5. Separate the fraction into a product of two reciprocal functions.

1/cos(x)⋅1/sin(x)

6. Substitute the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x) to reach the final form.

sec(x)*csc(x)

Final Answer

1/tan(x)+tan(x)=sec(x)*csc(x)

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