Step 1 :First, we need to isolate the cosine function. The equation is already in the form \(\cos \left(2 \theta-\frac{\pi}{2}\right)=1\), so we don't need to do anything here.
Step 2 :Next, we need to find the values of \(2 \theta-\frac{\pi}{2}\) that make the cosine function equal to 1. We know that \(\cos \theta = 1\) when \(\theta = 0\) or \(\theta = 2\pi\) in the interval \([0, 2\pi)\). So, we have \(2 \theta-\frac{\pi}{2} = 0\) or \(2 \theta-\frac{\pi}{2} = 2\pi\).
Step 3 :Solving the first equation \(2 \theta-\frac{\pi}{2} = 0\) for \(\theta\), we get \(\theta = \frac{\pi}{4}\).
Step 4 :Solving the second equation \(2 \theta-\frac{\pi}{2} = 2\pi\) for \(\theta\), we get \(\theta = \frac{5\pi}{4}\).
Step 5 :Therefore, the solutions to the equation \(\cos \left(2 \theta-\frac{\pi}{2}\right)=1\) in the interval \([0,2 \pi)\) are \(\theta = \frac{\pi}{4}, \frac{5\pi}{4}\).
Step 6 :Finally, we check our solutions. Substituting \(\theta = \frac{\pi}{4}\) into the original equation, we get \(\cos \left(2 \cdot \frac{\pi}{4}-\frac{\pi}{2}\right)=1\), which simplifies to \(1=1\). Substituting \(\theta = \frac{5\pi}{4}\) into the original equation, we get \(\cos \left(2 \cdot \frac{5\pi}{4}-\frac{\pi}{2}\right)=1\), which also simplifies to \(1=1\). Therefore, our solutions are correct.
Step 7 :So, the solutions to the equation \(\cos \left(2 \theta-\frac{\pi}{2}\right)=1\) in the interval \([0,2 \pi)\) are \(\boxed{\theta = \frac{\pi}{4}, \frac{5\pi}{4}}\).