Select one or more expressions that together represent all solutions to the equation. Your answer should be in radians.
Assume
\[n\] is any integer.
\[-4\cos(5x)+1=1\]
Choose all answers that apply:
Choose all answers that apply:
(Choice A)
\[-\pi+n\cdot2\pi\]
A
\[-\pi+n\cdot2\pi\]
(Choice B)

\[-\dfrac{\pi}{2}+n\cdot2\pi\]
B

\[-\dfrac{\pi}{2}+n\cdot2\pi\]
(Choice C)

\[-\dfrac{\pi}{10}+n\cdot\dfrac{2\pi}{5}\]
C

\[-\dfrac{\pi}{10}+n\cdot\dfrac{2\pi}{5}\]
(Choice D)

\[\dfrac{\pi}{20}+n\cdot\dfrac{\pi}{5}\]
D

\[\dfrac{\pi}{20}+n\cdot\dfrac{\pi}{5}\]
(Choice E)

\[\dfrac{\pi}{10}+n\cdot\dfrac{2\pi}{5}\]
E

\[\dfrac{\pi}{10}+n\cdot\dfrac{2\pi}{5}\]
(Choice F)

\[\dfrac{\pi}{2}+n\cdot\pi\]
F

\[\dfrac{\pi}{2}+n\cdot\pi\]

Let's start by solving the equation:

\[-4\cos(5x) + 1 = 1\]

First, isolate the cosine term:

\[-4\cos(5x) = 0\]

Divide both sides by \(-4\):

\[\cos(5x) = 0\]

Next, recall the general solution for \(\cos(\theta) = 0\), which is:

\[
\theta = \dfrac{\pi}{2} + n\pi \quad \text{where } n \text{ is any integer}
\]

Substitute \(\theta = 5x\):

\[
5x = \dfrac{\pi}{2} + n\pi
\]

Now, solve for \(x\):

\[
x = \dfrac{\pi}{10} + n\cdot\dfrac{\pi}{5}
\]

This corresponds to choice **E**:

\[
\boxed{\dfrac{\pi}{10} + n\cdot\dfrac{2\pi}{5}}
\]

To ensure we find all solutions, consider the periodicity of the cosine function. The general solution accounts for all values of \(x\) that satisfy the original equation. The other answer choices represent either incorrect periodicities or incorrect specific angles.

Thus, the correct and complete answer is:

- **E**: \(\dfrac{\pi}{10} + n\cdot\dfrac{2\pi}{5}\)

This is the only choice that correctly represents all solutions to the given equation.

Would you like any further explanation or details? 

Here are some related questions:
1. How would the solutions change if the equation were \( -4\cos(5x) + 1 = 2 \)?
2. What are the general solutions for \( \sin(kx) = 0 \) and \( \tan(kx) = 0 \)?
3. What is the significance of the coefficient in front of \( x \) in trigonometric equations?
4. How does the cosine function's period affect the solution set of trigonometric equations?
5. How can you verify if the solutions obtained are correct?

**Tip:** Always check if the equation can be simplified before isolating the trigonometric function for easier solving.

Learn how to solve the trigonometric equation -4cos(5x) + 1 = 1 step-by-step. Understand the general solution for cos(θ) = 0 and apply it to find all possible solutions in radians. Suitable for advanced high school students studying trigonometry.

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