Find the sum of the values of $x$ such that $\cos^{3}3x+\cos^{3}5x= 8\cos^{3}4x\cos^{3}x$, where $x$ is measured in degrees and $100 ^\circ < x < 200^\circ$.

We are asked to find the sum of the values of $ x $ such that 

\[
\cos^{3}(3x) + \cos^{3}(5x) = 8 \cos^{3}(4x) \cos^{3}(x)
\]

where $ x $ is measured in degrees and $ 100^\circ < x < 200^\circ $.

### Step 1: Simplify the equation
We start by rewriting the given equation:

\[
\cos^3(3x) + \cos^3(5x) = 8 \cos^3(4x) \cos^3(x)
\]

Using the identity for the sum of cubes, $ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $, with $ a = \cos(3x) $ and $ b = \cos(5x) $, we can express the left-hand side as:

\[
\cos^3(3x) + \cos^3(5x) = (\cos(3x) + \cos(5x)) \left(\cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x)\right)
\]

### Step 2: Use sum-to-product identities
To simplify further, use the sum-to-product identity:

\[
\cos(3x) + \cos(5x) = 2 \cos\left(\frac{3x + 5x}{2}\right) \cos\left(\frac{5x - 3x}{2}\right) = 2 \cos(4x) \cos(x)
\]

Thus, the equation simplifies to:

\[
2 \cos(4x) \cos(x) \left(\cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x)\right) = 8 \cos^3(4x) \cos^3(x)
\]

### Step 3: Factor and solve the equation
We can divide both sides by $ 2 \cos(4x) \cos(x) $, assuming $ \cos(4x) \cos(x) \neq 0 $:

\[
\left(\cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x)\right) = 4 \cos^2(4x) \cos^2(x)
\]

However, another potential solution occurs when $ 2 \cos(4x) \cos(x) = 0 $. This happens if either $ \cos(4x) = 0 $ or $ \cos(x) = 0 $.

### Step 4: Consider special cases
#### Case 1: $ \cos(4x) = 0 $
This occurs when:

\[
4x = 90^\circ + 180^\circ n \quad \text{for integer } n
\]

This simplifies to:

\[
x = 22.5^\circ + 45^\circ n
\]

We need $ x $ such that $ 100^\circ < x < 200^\circ $. The possible values are $ n = 2, 3, 4 $, giving $ x = 112.5^\circ, 157.5^\circ, 202.5^\circ $. Only $ x = 112.5^\circ $ and $ x = 157.5^\circ $ are within the given range.

#### Case 2: $ \cos(x) = 0 $
This occurs when:

\[
x = 90^\circ + 180^\circ n
\]

But $ x $ must lie between $ 100^\circ $ and $ 200^\circ $. Hence, the only solution is $ x = 180^\circ $.

### Step 5: Sum of values
The valid solutions within the specified range are $ x = 112.5^\circ, 157.5^\circ, $ and $ 180^\circ $. Therefore, the sum of the values is:

\[
112.5^\circ + 157.5^\circ + 180^\circ = 450^\circ
\]

So, the sum of the values of $ x $ is $ \boxed{450^\circ} $.

Do you have any questions, or would you like further details?

Here are five related questions you might find helpful:
1. How can sum-to-product identities be used to simplify trigonometric equations?
2. What is the significance of the sum of cubes identity in trigonometry?
3. Can the process used here apply to different ranges for $ x $?
4. How does the cosine function behave across different quadrants?
5. What are other trigonometric identities that are useful for solving such equations?

**Tip:** When solving trigonometric equations, always check the range and consider special cases where certain trigonometric functions could be zero.

Discover how to find the sum of values of x satisfying the trigonometric equation cos^3(3x) + cos^3(5x) = 8 cos^3(4x) cos^3(x), within the range 100° < x < 200°. Detailed step-by-step solution provided.

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