Verify the Identity sec(x)-tan(x)sin(x)=1/(sec(x))

Problem

sec(x)−tan(x)*sin(x)=1/sec(x)

Solution

1. Rewrite the terms on the left side using sine and cosine by applying the identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

sec(x)−tan(x)*sin(x)=1/cos(x)−sin(x)/cos(x)⋅sin(x)

2. Multiply the terms in the second part of the expression to combine the sine functions.

1/cos(x)−sin2(x)/cos(x)

3. Combine the fractions since they share a common denominator of cos(x)

(1−sin2(x))/cos(x)

4. Apply the Pythagorean identity cos2(x)+sin2(x)=1 which implies 1−sin2(x)=cos2(x)

cos2(x)/cos(x)

5. Simplify the fraction by dividing the numerator by the denominator.

cos(x)

6. Relate the result to the right side of the original identity using the reciprocal identity cos(x)=1/sec(x)

cos(x)=1/sec(x)

Final Answer

sec(x)−tan(x)*sin(x)=1/sec(x)

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