Simplify (tan(x))/(csc(x))+(sin(x))/(tan(x))

Problem

tan(x)/csc(x)+sin(x)/tan(x)

Solution

1. Rewrite the trigonometric functions in terms of sin(x) and cos(x) using the identities tan(x)=sin(x)/cos(x) csc(x)=1/sin(x) and 1/tan(x)=cos(x)/sin(x)

sin(x)/cos(x)/1/sin(x)+sin(x)/sin(x)/cos(x)

2. Simplify each complex fraction by multiplying the numerator by the reciprocal of the denominator.

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

3. Perform the multiplication in both terms.

sin2(x)/cos(x)+cos(x)

4. Find a common denominator to combine the terms.

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

5. Combine the numerators over the common denominator.

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

6. Apply the Pythagorean identity sin2(x)+cos2(x)=1

1/cos(x)

7. Use the reciprocal identity 1/cos(x)=sec(x)

sec(x)

Final Answer

tan(x)/csc(x)+sin(x)/tan(x)=sec(x)

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