Verify the Identity sin(t)tan(t)=(1-cos(t)^2)/(cos(t))

Problem

sin(t)*tan(t)=(1−cos2(t))/cos(t)

Solution

1. Identify the left side of the identity to be transformed.

sin(t)*tan(t)

2. Substitute the quotient identity for tangent, which is tan(t)=sin(t)/cos(t)

sin(t)⋅sin(t)/cos(t)

3. Multiply the terms to combine them into a single fraction.

sin2(t)/cos(t)

4. Apply the Pythagorean identity sin2(t)+cos2(t)=1 which can be rearranged to sin2(t)=1−cos2(t)

(1−cos2(t))/cos(t)

5. Compare the result to the right side of the original equation to confirm they are identical.

(1−cos2(t))/cos(t)=(1−cos2(t))/cos(t)

Final Answer

sin(t)*tan(t)=(1−cos2(t))/cos(t)

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