Verify the Identity sin(x)tan(x)=sec(x)-cos(x)

Problem

sin(x)*tan(x)=sec(x)−cos(x)

Solution

1. Rewrite the tangent function on the left side using the quotient identity tan(x)=sin(x)/cos(x)

sin(x)⋅sin(x)/cos(x)

2. Multiply the terms to combine them into a single fraction.

sin2(x)/cos(x)

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

(1−cos2(x))/cos(x)

4. Split the fraction into two separate terms by dividing each part of the numerator by the denominator.

1/cos(x)−cos2(x)/cos(x)

5. Simplify each term using the reciprocal identity sec(x)=1/cos(x) and basic division.

sec(x)−cos(x)

Final Answer

sin(x)*tan(x)=sec(x)−cos(x)

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