QuestionFind all solutions of the equation in the interval $[0,2 \pi)$. $$-\sqrt{1-\tan x}=\sec x$$
Write your answer(s) in radians in terms of $\pi$. If there is more than one solution, separate them with commas. $$x=$$ $\square$

Studdy Solution
Check each solution to ensure it satisfies the original equation $-\sqrt{1-\tan x} = \sec x$.
1. $x = 0$: $-\sqrt{1-0} = -1$, $\sec 0 = 1$. Does not satisfy.
2. $x = \pi$: $-\sqrt{1-0} = -1$, $\sec \pi = -1$. Satisfies.
3. $x = \frac{3\pi}{4}$: $-\sqrt{1-(-1)} = -\sqrt{2}$, $\sec \frac{3\pi}{4} = -\sqrt{2}$. Satisfies.
4. $x = \frac{7\pi}{4}$: $-\sqrt{1-(-1)} = -\sqrt{2}$, $\sec \frac{7\pi}{4} = \sqrt{2}$. Does not satisfy.
The solutions in the interval $[0, 2\pi)$ are:
$$x = \pi, \frac{3\pi}{4}$$