Use trigonometric identities and \&Agebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in $[0,2 \pi)$ and indicating the remaining answers by using $n$ to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution." \[ \sec ^{2}(x)+5=7 \]

Step 1 ：The given equation is \(\sec ^{2}(x)+5=7\). Subtract 5 from both sides of the equation to isolate \(\sec ^{2}(x)\) on one side, giving \(\sec ^{2}(x) = 2\).

Step 2 ：Since \(\sec(x)\) is the reciprocal of \(\cos(x)\), rewrite the equation as \(\cos ^{2}(x) = 1/2\).

Step 3 ：Take the square root of both sides to get \(\cos(x) = \pm \sqrt{1/2}\).

Step 4 ：Find the values of \(x\) in the interval \([0, 2\pi)\) that satisfy this equation. The solutions are approximately \(0.7854\) and \(5.4978\).

Step 5 ：Since the cosine function has a period of \(2\pi\), the remaining solutions can be found by adding any integer multiple of \(2\pi\) to these solutions.

Step 6 ：\(\boxed{\text{The solutions to the equation } \sec ^{2}(x)+5=7 \text{ in the interval } [0, 2\pi) \text{ are approximately } 0.7854 \text{ and } 5.4978. \text{ The remaining solutions can be found by adding any integer multiple of } 2\pi \text{ to these solutions. Therefore, the solutions are } x \approx 0.7854 + 2n\pi \text{ and } x \approx 5.4978 + 2n\pi, \text{ where } n \text{ is any integer.}}\)