expand 8(cos*x*)^4 using trigonometry function

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 +

Discover how to expand 8(cos x)^4 using trigonometric functions with a detailed step-by-step solution. This problem covers trigonometric identities, specifically the double-angle identity for cosine. Suitable for advanced high school students.

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