4cos2-t-1=0

To solve the equation \( 4\cos(2t) - 1 = 0 \), follow these steps:

1. Isolate the cosine term: \[ 4\cos(2t) = 1 \] \[ \cos(2t) = \frac{1}{4} \]

2. Find the general solution for \( 2t \): The cosine function has solutions in the form: \[ 2t = \cos^{-1}\left(\frac{1}{4}\right) + 2k\pi \quad \text{and} \quad 2t = -\cos^{-1}\left(\frac{1}{4}\right) + 2k\pi \] where \( k \) is any integer.

3. Solve for \( t \): Dividing both sides by 2 gives: \[ t = \frac{1}{2} \cos^{-1}\left(\frac{1}{4}\right) + k\pi \quad \text{and} \quad t = -\frac{1}{2} \cos^{-1}\left(\frac{1}{4}\right) + k\pi \]

This gives you the general solutions for the original equation. You can calculate \( \cos^{-1}\left(\frac{1}{4}\right) \) using a calculator for a numerical value if needed.