Simplify (1-cos(x))(1+sec(x))(cos(x))

Problem

(1−cos(x))*(1+sec(x))*(cos(x))

Solution

1. Distribute the cos(x) term into the second binomial (1+sec(x))

(1−cos(x))*(cos(x)⋅1+cos(x)⋅sec(x))

2. Apply the reciprocal identity sec(x)=1/cos(x) to simplify the second binomial.

(1−cos(x))*(cos(x)+cos(x)⋅1/cos(x))

(1−cos(x))*(cos(x)+1)

3. Rearrange the terms in the second binomial to recognize a common algebraic pattern.

(1−cos(x))*(1+cos(x))

4. Apply the difference of squares formula (a−b)*(a+b)=a2−b2

1−cos2(x)

1−cos2(x)

5. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−cos2(x)=sin2(x)

sin2(x)

Final Answer

(1−cos(x))*(1+sec(x))*(cos(x))=sin2(x)

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