Solve for the exact solutions in the interval $[0,2 \pi)$. If the equation has no solutions, respond with DNE. $2 \sec ^{2}(x)=3-\tan (x)$ Calculator

Step 1 ：\(2\sec^2(x) = 3 - \tan(x)\)

Step 2 ：Rewrite \(\sec^2(x)\) as \(1 + \tan^2(x)\)

Step 3 ：\(2(1 + \tan^2(x)) = 3 - \tan(x)\)

Step 4 ：Simplify to get \(2\tan^2(x) - \tan(x) + 2 = 3\)

Step 5 ：Subtract 3 from both sides to get \(2\tan^2(x) - \tan(x) - 1 = 0\)

Step 6 ：This is a quadratic equation in terms of \(\tan(x)\). Solve it using the quadratic formula

Step 7 ：\(\tan(x) = \frac{-(-1) \pm \sqrt{(-1)^2 - 4*2*(-1)}}{2*2}\)

Step 8 ：Solving this gives \(\tan(x) = \frac{1 \pm \sqrt{9}}{4}\)

Step 9 ：So, \(\tan(x) = \frac{1 \pm 3}{4}\)

Step 10 ：This gives two possible solutions for \(\tan(x)\): \(\tan(x) = 1\) and \(\tan(x) = -0.5\)

Step 11 ：For \(\tan(x) = 1\), \(x = \frac{\pi}{4}\) and \(x = \frac{5\pi}{4}\)

Step 12 ：For \(\tan(x) = -0.5\), \(x = \arctan(-0.5)\) and \(x = \pi + \arctan(-0.5)\)

Step 13 ：Using a calculator, we find that \(\arctan(-0.5) \approx -0.4636\). Adding \(\pi\) to this gives \(\approx 2.6779\)

Step 14 ：\(\boxed{x = \frac{\pi}{4}, x = \frac{5\pi}{4}, x \approx -0.4636, x \approx 2.6779}\)