To verify the given trigonometric identity:
[tex]\[
\sin^4 \theta = \frac{1}{8}(3 - 4 \cos 2\theta + \cos 4\theta),
\][/tex]
we will follow a detailed, step-by-step approach to establish that both sides of the equation are indeed equal.
### Step 1: Expand [tex]\(\sin^4 \theta\)[/tex]
We start by expressing [tex]\(\sin^4 \theta\)[/tex] in terms of standard trigonometric identities. Recall the power-reduction formula for [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\sin^2 \theta = \frac{1 - \cos 2\theta}{2}.
\][/tex]
Then, squaring [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\sin^4 \theta = \left( \sin^2 \theta \right)^2 = \left( \frac{1 - \cos 2\theta}{2} \right)^2 = \frac{(1 - \cos 2\theta)^2}{4}.
\][/tex]
### Step 2: Expand the expression [tex]\((1 - \cos 2\theta)^2\)[/tex]
Next, expand the square:
[tex]\[
(1 - \cos 2\theta)^2 = 1 - 2\cos 2\theta + \cos^2 2\theta.
\][/tex]
### Step 3: Utilize the power-reduction formula for [tex]\(\cos^2 2\theta\)[/tex]
We can use another power-reduction formula for [tex]\(\cos^2 2\theta\)[/tex]:
[tex]\[
\cos^2 2\theta = \frac{1 + \cos 4\theta}{2}.
\][/tex]
### Step 4: Substitute [tex]\(\cos^2 2\theta\)[/tex] into the expression
Substitute this back into the expanded expression:
[tex]\[
\sin^4 \theta = \frac{1 - 2\cos 2\theta + \frac{1 + \cos 4\theta}{2}}{4}.
\][/tex]
Simplify the combined terms:
[tex]\[
= \frac{2(1 - 2\cos 2\theta) + 1 + \cos 4\theta}{8}
= \frac{2 - 4\cos 2\theta + 1 + \cos 4\theta}{8}
= \frac{3 - 4\cos 2\theta + \cos 4\theta}{8}.
\][/tex]
### Step 5: Compare with the right-hand side
Now we see that the expression we obtained matches exactly with the given right-hand side of the original identity:
[tex]\[
\frac{1}{8}(3 - 4 \cos 2 \theta + \cos 4 \theta).
\][/tex]
Since both sides are equal, we have verified the trigonometric identity:
[tex]\[
\sin^4 \theta = \frac{1}{8}(3 - 4 \cos 2 \theta + \cos 4 \theta).
\][/tex]
Thus, the identity is confirmed to be true.
