Verify the Identity (tan(t)^2)/(sec(t))=sec(t)-cos(t)

Problem

tan2(t)/sec(t)=sec(t)−cos(t)

Solution

1. Identify the Pythagorean identity for the numerator on the left side.

tan2(t)=sec2(t)−1

2. Substitute the identity into the expression.

(sec2(t)−1)/sec(t)

3. Split the fraction into two separate terms by dividing each term in the numerator by the denominator.

sec2(t)/sec(t)−1/sec(t)

4. Simplify the first term by canceling common factors and apply the reciprocal identity cos(t)=1/sec(t) to the second term.

sec(t)−cos(t)

Final Answer

tan2(t)/sec(t)=sec(t)−cos(t)

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