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

Problem

1−2*cos^2(x)+cos^4(x)

Solution

1. Identify the expression as a quadratic form by letting u=cos^2(x)

2. Rewrite the expression in terms of u to see the structure of a perfect square trinomial.

u^2−2*u+1

3. Factor the quadratic expression using the pattern a^2−2*a*b+b^2=(a−b)^2

(u−1)^2

4. Substitute the original term cos^2(x) back in for u

(cos^2(x)−1)^2

5. Apply the difference of squares formula a^2−b^2=(a−b)*(a+b) to the expression inside the parentheses.

((cos(x)−1)*(cos(x)+1))^2

6. Distribute the exponent to both factors to reach the fully factored form.

(cos(x)−1)^2*(cos(x)+1)^2

Final Answer

1−2*cos^2(x)+cos^4(x)=(cos(x)−1)^2*(cos(x)+1)^2

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