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

Problem

sec(x)−sin(x)*tan(x)=cos(x)

Solution

1. Rewrite the expression using basic trigonometric identities to express all terms in terms of sine and cosine.

sec(x)=1/cos(x)

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

2. Substitute these identities into the left side of the equation.

1/cos(x)−sin(x)⋅sin(x)/cos(x)

3. Multiply the terms in the second part of the expression.

1/cos(x)−sin2(x)/cos(x)

4. Combine the fractions since they share a common denominator.

(1−sin2(x))/cos(x)

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

cos2(x)/cos(x)

6. Simplify the fraction by canceling one factor of cos(x) from the numerator and denominator.

cos(x)

Final Answer

sec(x)−sin(x)*tan(x)=cos(x)

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