To expand $8(\cos x)^4$ using trigonometric functions, we can follow a similar approach as we did with $(\sin x)^4$.

First, let's use the double-angle identity for cosine:

$\cos^2 x = \frac{1 + \cos 2x}{2}$

Now, we want to find $\cos^4 x$, which can be done by squaring $\cos^2 x$:

$\cos^4 x = \left( \frac{1 + \cos 2x}{2} \right)^2 = \frac{(1 + \cos 2x)^2}{4}$

Expand the square in the numerator:

$(1 + \cos 2x)^2 = 1 + 2\cos 2x + \cos^2 2x$

Therefore:

$\cos^4 x = \frac{1 + 2\cos 2x + \cos^2 2x}{4}$

Now, substitute this back into the original expression:

$8(\cos x)^4 = 8 \cdot \frac{1 + 2\cos 2x + \cos^2 2x}{4}$

Simplify by multiplying both the numerator and the denominator by 2:

$8(\cos x)^4 = 2(1 + 2\cos 2x + \cos^2 2x)$

Distribute the 2:

$8(\cos x)^4 = 2 + 4\cos 2x + 2\cos^2 2x$

So, the expanded form of $8(\cos x)^4$ using trigonometric functions is:

$8(\cos x)^4 = 2 + 4\cos 2x + 2\cos^2 2x$

Would you like further details or have any specific questions about this transformation?

Here are some relative questions that expand on this information:

1. How would you graph the function $f(x) = 2 + 4\cos 2x + 2\cos^2 2x$?
2. What is the derivative of $2 + 4\cos 2x + 2\cos^2 2x$ with respect to $x$?
3. How does the expression $2 + 4\cos 2x + 2\cos^2 2x$ change if we replace $x$ with $x/2$?
4. What are the critical points of the function $f(x) = 2 + 4\cos 2x + 2\cos^2 2x$?
5. Can you express $2 + 4\cos 2x + 2\cos^2 2x$ in a simplified form using other trigonometric identities?
6. What is the integral of ( 2 + 4\cos 2x +