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

Problem

1−2*sin^2(x)+sin^4(x)

Solution

1. Rearrange the expression in descending order of powers to make the structure of the trinomial clearer.

sin^4(x)−2*sin^2(x)+1

2. Rewrite the middle term to prepare for grouping by splitting −2*sin^2(x) into two identical terms.

sin^4(x)−sin^2(x)−sin^2(x)+1

3. Group the first two terms and the last two terms together.

(sin^4(x)−sin^2(x))+(−sin^2(x)+1)

4. Factor out the greatest common factor from each group, which is sin^2(x) from the first group and −1 from the second group.

sin^2(x)*(sin^2(x)−1)−1*(sin^2(x)−1)

5. Factor out the common binomial (sin^2(x)−1)

(sin^2(x)−1)*(sin^2(x)−1)

6. Simplify the expression by writing it as a squared binomial.

(sin^2(x)−1)^2

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

((sin(x)−1)*(sin(x)+1))^2

8. Distribute the exponent to both factors.

(sin(x)−1)^2*(sin(x)+1)^2

Final Answer

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

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