Questions: cos x tan x - cos x = 0

△ Solve the trigonometric equation \(\cos x \tan x - \cos x = 0\). ○ Factor the equation ☼ The equation \(\cos x (\tan x - 1) = 0\) is satisfied if \(\cos x = 0\) or \(\tan x = 1\). ○ Solve \(\cos x = 0\) ☼ If \(\cos x = 0\), then \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer. ○ Solve \(\tan x = 1\) ☼ If \(\tan x = 1\), then \(x = \frac{\pi}{4} + n\pi\), which can be rewritten as \(x = \frac{\pi}{4} + 2n\pi\) or \(x = \frac{5\pi}{4} + 2n\pi\). ✧ The solutions are \(x = \frac{\pi}{2} + n\pi\), \(x = \frac{\pi}{4} + 2n\pi\), or \(x = \frac{5\pi}{4} + 2n\pi\).

△ Determine the correct option from the given choices. ○ Compare solutions with options ☼ The correct solutions are \(x = \frac{\pi}{2} + n\pi\) and either \(x = \frac{\pi}{4} + 2n\pi\) or \(x = \frac{5\pi}{4} + 2n\pi\). Options A and D match these solutions. ✧ Both options A (\(\frac{\pi}{2}+n \pi, \frac{5 \pi}{4}+2 n \pi\)) and D (\(\frac{\pi}{2}+n \pi, \frac{\pi}{4}+2 n \pi\)) are correct. Option D is selected. ☺ $\frac{\pi}{2}+n \pi, \frac{\pi}{4}+2 n \pi$ or $\frac{\pi}{2}+n \pi, \frac{5 \pi}{4}+2 n \pi$ The answer is A or D. Since I can only pick one, I will pick D.