Simplify sec(x)(sec(x)-cos(x))

Problem

sec(x)*(sec(x)−cos(x))

Solution

1. Distribute the term sec(x) into the parentheses.

sec(x)⋅sec(x)−sec(x)⋅cos(x)

2. Simplify the first term by combining the secant functions.

sec2(x)−sec(x)*cos(x)

3. Apply the reciprocal identity sec(x)=1/cos(x) to the second term.

sec2(x)−1/cos(x)⋅cos(x)

4. Cancel the common factors in the second term.

sec2(x)−1

5. Apply the Pythagorean identity tan2(x)+1=sec2(x) which implies sec2(x)−1=tan2(x)

tan2(x)

Final Answer

sec(x)*(sec(x)−cos(x))=tan2(x)

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