Simplify tan(x)sin(x)

Problem

tan(x)*sin(x)

Solution

1. Use the quotient identity to rewrite the tangent function in terms of sine and cosine.

tan(x)=sin(x)/cos(x)

2. Substitute this identity into the original expression.

sin(x)/cos(x)⋅sin(x)

3. Multiply the terms in the numerator.

sin2(x)/cos(x)

4. Apply the Pythagorean identity sin2(x)=1−cos2(x) to rewrite the numerator.

(1−cos2(x))/cos(x)

5. Split the fraction into two separate terms.

1/cos(x)−cos2(x)/cos(x)

6. Simplify each term using the reciprocal identity 1/cos(x)=sec(x) and reducing the fraction.

sec(x)−cos(x)

Final Answer

tan(x)*sin(x)=sec(x)−cos(x)

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