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

Problem

1−2*cos2(x)+cos4(x)

Solution

1. Identify the expression as a quadratic form by letting u=cos2(x)

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

u2−2*u+1

3. Factor the quadratic expression using the pattern a2−2*a*b+b2=(a−b)2

(u−1)2

4. Substitute the original term cos2(x) back in for u

(cos2(x)−1)2

5. Apply the difference of squares formula a2−b2=(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*cos2(x)+cos4(x)=(cos(x)−1)2*(cos(x)+1)2

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