Questions: Find all solutions to the equation (4 tan ^3(x)-4 tan (x)=0) over the interval (0, pi)). A) (x=0, x=fracpi4) B) (x=0, x=fracpi4, x=frac3 pi4) C) (x=0, x=fracpi4, x=frac5 pi4) D) (x=0, x=fracpi4, x=frac-pi4) E) (x=0, x=fracpi4, x=fracpi2) F) None of the above

Solution Steps

To solve the equation \(4 \tan^3(x) - 4 \tan(x) = 0\) over the interval \([0, \pi)\), we can factor the equation and solve for \(\tan(x)\). The factored form is \(4 \tan(x) (\tan^2(x) - 1) = 0\). This gives us two equations to solve: \(\tan(x) = 0\) and \(\tan^2(x) = 1\). We then find the values of \(x\) that satisfy these equations within the given interval.

We start with the equation

\[ 4 \tan^3(x) - 4 \tan(x) = 0. \]

Factoring out \(4 \tan(x)\), we have

\[ 4 \tan(x) (\tan^2(x) - 1) = 0. \]

This gives us two cases to solve:

1. \( \tan(x) = 0 \)
2. \( \tan^2(x) - 1 = 0 \) which simplifies to \( \tan(x) = \pm 1 \).

Case 1: For \( \tan(x) = 0 \), the solutions in the interval \([0, \pi)\) are

\[ x = 0. \]

Case 2: For \( \tan(x) = 1 \), we find

\[ x = \frac{\pi}{4}. \]

For \( \tan(x) = -1 \), the solution in the interval \([0, \pi)\) is

\[ x = \frac{3\pi}{4}. \]

The complete set of solutions in the interval \([0, \pi)\) is

\[ x = 0, \quad x = \frac{\pi}{4}, \quad x = \frac{3\pi}{4}. \]

Now we compare our solutions with the provided options:

• A) \(x = 0, x = \frac{\pi}{4}\)
• B) \(x = 0, x = \frac{\pi}{4}, x = \frac{3\pi}{4}\)
• C) \(x = 0, x = \frac{\pi}{4}, x = \frac{5\pi}{4}\)
• D) \(x = 0, x = \frac{\pi}{4}, x = -\frac{\pi}{4}\)
• E) \(x = 0, x = \frac{\pi}{4}, x = \frac{\pi}{2}\)

The correct option that includes all solutions is

\[ \text{Option B: } x = 0, x = \frac{\pi}{4}, x = \frac{3\pi}{4}. \]

Final Answer