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

Problem

tan(x)*(csc(x)−sin(x))=cos(x)

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the identities tan(x)=sin(x)/cos(x) and csc(x)=1/sin(x)

tan(x)*(csc(x)−sin(x))=sin(x)/cos(x)*(1/sin(x)−sin(x))

2. Distribute the term sin(x)/cos(x) into the parentheses.

sin(x)/cos(x)⋅1/sin(x)−sin(x)/cos(x)⋅sin(x)

3. Simplify the products by canceling sin(x) in the first term and multiplying in the second term.

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 the numerator.

cos2(x)/cos(x)

6. Divide the terms to reach the final simplified form.

cos(x)

Final Answer

tan(x)*(csc(x)−sin(x))=cos(x)

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