Question

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

Answer 1

`\left(1 - \sin^(2)(x)\right)^(2)`.

Explanation:

Notice that "
`1`" is equal to "
`1^(2)`" whereas
`\sin^(4)(x) = {\left(\sin^(2)(x)\right)}^(2)`. Therefore:

`\begin{aligned} & 1 - 2\, \sin^(2)(x) + \sin^(4)(x) \\ =\; & 1^(2) - 2\, \sin^(2)(x) + {\left(\sin^(2)(x)\right)}^(2)\end{aligned}`.

Make use of the identity
`(a - b)^(2) = a^(2) - 2\, a\, b + b^(2)`. In this case, set
`a = 1` whereas
`b = \sin^(2)(x)`. Therefore:

`\begin{aligned} & 1^(2) - 2\, \sin^(2)(x) + {\left(\sin^(2)(x)\right)}^(2) \\ =\; & a^(2) - 2\, a\, b + b^(2) \\ =\; & (a - b)^(2) \\ =\; & {\left(1 - \sin^(2)(x)\right)}^(2)\end{aligned}`.