Verify the Identity tan(a)+cot(a)=sec(a)*csc(a)

Problem

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

Solution

1. Rewrite the left side of the equation in terms of sine and cosine using the definitions tan(a)=sin(a)/cos(a) and cot(a)=cos(a)/sin(a)

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

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

(sin(a)*sin(a))/(cos(a)*sin(a))+(cos(a)*cos(a))/(cos(a)*sin(a))

3. Simplify the numerator by combining the terms over the common denominator.

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

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

1/(cos(a)*sin(a))

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

1/cos(a)⋅1/sin(a)

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

sec(a)*csc(a)

Final Answer

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

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