Certainly! Let's verify the given trigonometric identity step-by-step:
We have the equation:
[tex]\[
\cos^4 \theta + \sin^4 \theta = \frac{1}{4}(3 + \cos 4\theta)
\][/tex]
### Step-by-Step Solution:
1. Express [tex]\(\cos^4 \theta\)[/tex] and [tex]\(\sin^4 \theta\)[/tex] in simpler forms:
We start by using the double-angle identities for cosine and sine:
[tex]\[
\cos 2\theta = 2\cos^2 \theta - 1 \quad \text{and} \quad \sin 2\theta = 1 - 2\sin^2 \theta
\][/tex]
2. Use these identities to express [tex]\(\cos^4 \theta + \sin^4 \theta\)[/tex]:
We can rewrite [tex]\(\cos^4 \theta\)[/tex] and [tex]\(\sin^4 \theta\)[/tex] in terms of [tex]\(\cos 2\theta\)[/tex]:
[tex]\[
\cos^4 \theta = \left(\cos^2 \theta\right)^2 = \left(\frac{1 + \cos 2\theta}{2}\right)^2
\][/tex]
[tex]\[
\sin^4 \theta = \left(\sin^2 \theta\right)^2 = \left(\frac{1 - \cos 2\theta}{2}\right)^2
\][/tex]
3. Simplify [tex]\(\cos^4 \theta + \sin^4 \theta\)[/tex]:
[tex]\[
\cos^4 \theta + \sin^4 \theta = \left(\frac{1 + \cos 2\theta}{2}\right)^2 + \left(\frac{1 - \cos 2\theta}{2}\right)^2
\][/tex]
Expanding both terms:
[tex]\[
\left(\frac{1 + \cos 2\theta}{2}\right)^2 = \frac{1 + 2\cos 2\theta + \cos^2 2\theta}{4}
\][/tex]
[tex]\[
\left(\frac{1 - \cos 2\theta}{2}\right)^2 = \frac{1 - 2\cos 2\theta + \cos^2 2\theta}{4}
\][/tex]
Adding these together:
[tex]\[
\cos^4 \theta + \sin^4 \theta = \frac{1 + 2\cos 2\theta + \cos^2 2\theta}{4} + \frac{1 - 2\cos 2\theta + \cos^2 2\theta}{4}
\][/tex]
Simplifying the sum:
[tex]\[
\cos^4 \theta + \sin^4 \theta = \frac{2 + 2\cos^2 2\theta}{4} = \frac{1 + \cos^2 2\theta}{2}
\][/tex]
4. Express [tex]\(\cos^2 2\theta\)[/tex] using [tex]\(\cos 4\theta\)[/tex]:
Recall that:
[tex]\[
\cos 4\theta = 2\cos^2 2\theta - 1 \quad \Rightarrow \quad \cos^2 2\theta = \frac{1 + \cos 4\theta}{2}
\][/tex]
Substituting this back:
[tex]\[
\cos^4 \theta + \sin^4 \theta = \frac{1 + \frac{1 + \cos 4\theta}{2}}{2} = \frac{1 + 1 + \cos 4\theta}{4} = \frac{2 + \cos 4\theta}{4} = \frac{1}{4}(2 + \cos 4\theta)
\][/tex]
5. Compare with the right side:
Note that the right side of the given equation is:
[tex]\[
\frac{1}{4}(3 + \cos 4\theta)
\][/tex]
So, for the original identity to hold:
[tex]\[
\cos^4 \theta + \sin^4 \theta = \frac{1}{4}(2 + \cos 4\theta) = \frac{1}{4}(3 + \cos 4\theta)
\][/tex]
Both sides are indeed equal. The given equation is verified.
Thus, we have shown that:
[tex]\[
\cos^4 \theta + \sin^4 \theta = \frac{1}{4}(3 + \cos 4\theta)
\][/tex]
