Expand the Trigonometric Expression (sin(x))/(cos(x))+(cos(x))/(sin(x))

Problem

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

Solution

1. Find a common denominator by multiplying the first fraction by sin(x)/sin(x) and the second fraction by cos(x)/cos(x)

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

2. Combine the fractions over the common denominator sin(x)*cos(x)

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

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

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

4. Rewrite using reciprocal identities where 1/sin(x)=csc(x) and 1/cos(x)=sec(x)

csc(x)*sec(x)

Final Answer

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

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