Simplify cos(t)(sec(t)-cos(t))

Problem

cos(t)*(sec(t)−cos(t))

Solution

1. Distribute the cos(t) term to both terms inside the parentheses.

cos(t)*sec(t)−cos2(t)

2. Apply the reciprocal identity for the secant function, where sec(t)=1/cos(t)

cos(t)⋅1/cos(t)−cos2(t)

3. Simplify the first term by canceling cos(t)

1−cos2(t)

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

sin2(t)

Final Answer

cos(t)*(sec(t)−cos(t))=sin2(t)

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