Simplify 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 in terms of cos^2(x)

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

3. Substitute a=1 and b=cos^2(x) into the perfect square formula.

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

5. Apply the Pythagorean identity sin^2(x)=1−cos^2(x) to simplify the term inside the parentheses.

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

Final Answer

1−2*cos^2(x)+cos^4(x)=sin^4(x)

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