Factor by Grouping 1-2cos(x)^2+cos(x)^4

Problem

1−2*cos(x)+cos(x)

Solution

1. Rearrange the expression in descending order of powers to make it easier to recognize the quadratic structure.

cos(x)−2*cos(x)+1

2. Substitute a variable to simplify the expression, letting u=cos(x)

u2−2*u+1

3. Split the middle term to prepare for grouping.

u2−u−u+1

4. Group the first two terms and the last two terms.

(u2−u)−(u−1)

5. Factor out the greatest common factor from each group.

u*(u−1)−1*(u−1)

6. Factor out the common binomial (u−1)

(u−1)*(u−1)

7. Rewrite as a square.

(u−1)2

8. Substitute back the original expression cos(x) for u

(cos(x)−1)2

9. Factor the difference of squares inside the parentheses, noting that cos(x)−1=(cos(x)−1)*(cos(x)+1)

((cos(x)−1)*(cos(x)+1))2

10. Distribute the exponent to both factors.

(cos(x)−1)2*(cos(x)+1)2

Final Answer

1−2*cos(x)+cos(x)=(cos(x)−1)2*(cos(x)+1)2

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