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

Problem

sec(x)−cos(x)=sin(x)*tan(x)

Solution

1. Express the left side in terms of sine and cosine by using the reciprocal identity sec(x)=1/cos(x)

sec(x)−cos(x)=1/cos(x)−cos(x)

2. Find a common denominator to combine the terms into a single fraction.

1/cos(x)−cos2(x)/cos(x)

(1−cos2(x))/cos(x)

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

sin2(x)/cos(x)

4. Rewrite the expression as a product of two trigonometric functions to match the right side of the identity.

sin(x)⋅sin(x)/cos(x)

5. Substitute the quotient identity sin(x)/cos(x)=tan(x) to complete the verification.

sin(x)*tan(x)

Final Answer

sec(x)−cos(x)=sin(x)*tan(x)

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