To solve the problem of finding which expression is equal to [tex]\( 1 - \cos^4 \theta \)[/tex], let us start by breaking down the expression and simplifying it step-by-step.
1. Expression Parsing:
We start with the function [tex]\( 1 - \cos^4 \theta \)[/tex].
2. Use the Pythagorean Identity:
Recall the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
From this, we can express [tex]\(\cos^2 \theta\)[/tex] as:
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta
\][/tex]
3. Square the Cosine Expression:
Next, let’s square [tex]\( \cos^2 \theta \)[/tex] to get [tex]\(\cos^4 \theta \)[/tex]:
[tex]\[
\cos^4 \theta = (1 - \sin^2 \theta)^2
\][/tex]
4. Expand the Squared Expression:
Now, let's expand [tex]\( (1 - \sin^2 \theta)^2 \)[/tex]:
[tex]\[
(1 - \sin^2 \theta)^2 = 1 - 2\sin^2 \theta + \sin^4 \theta
\][/tex]
5. Substitute Back into the Original Expression:
Substitute the expanded form back into the original expression [tex]\( 1 - \cos^4 \theta \)[/tex]:
[tex]\[
1 - \cos^4 \theta = 1 - (1 - 2\sin^2 \theta + \sin^4 \theta)
\][/tex]
6. Simplify the Expression:
Simplify the expression by distributing the negative sign and combining like terms:
[tex]\[
1 - (1 - 2\sin^2 \theta + \sin^4 \theta) = 1 - 1 + 2\sin^2 \theta - \sin^4 \theta
\][/tex]
[tex]\[
= 2\sin^2 \theta - \sin^4 \theta
\][/tex]
Therefore, the expression that is equal to [tex]\( 1 - \cos^4 \theta \)[/tex] is:
[tex]\[
2\sin^2 \theta - \sin^4 \theta
\][/tex]
So, the correct option is:
[tex]\[
\boxed{2 \sin^2 \theta - \sin^4 \theta}
\][/tex]
