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

Problem

1−2*cos2(x)+cos4(x)

Solution

1. Identify the expression as a quadratic form in terms of cos2(x)

2. Recognize the pattern of a perfect square trinomial, which follows the form a2−2*a*b+b2=(a−b)2

3. Substitute a=1 and b=cos2(x) into the perfect square formula.

4. Rewrite the expression as (1−cos2(x))2

5. Apply the Pythagorean identity sin2(x)=1−cos2(x) to simplify the term inside the parentheses.

6. Simplify the resulting expression (sin2(x))2 by multiplying the exponents.

Final Answer

1−2*cos2(x)+cos4(x)=sin4(x)

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