10. -/1 points details my notes larpcalclim4hs 5.3.030. find all solutions of the equation in the interval…

Explanation:

Step1: Rewrite the equation

Given (2\cos x + 2\sin x\tan x=4). Since (\tan x=\frac{\sin x}{\cos x}), the equation becomes (2\cos x + 2\sin x\times\frac{\sin x}{\cos x}=4). Multiply through by (\cos x) (assuming (\cos x\neq0)) to get (2\cos^{2}x + 2\sin^{2}x = 4\cos x). Using the identity (\sin^{2}x+\cos^{2}x = 1), the left - hand side is (2(\sin^{2}x+\cos^{2}x)=2). So the equation is (2 = 4\cos x).

Step2: Solve for (\cos x)

Divide both sides of (2 = 4\cos x) by 4, we have (\cos x=\frac{1}{2}).

Step3: Find solutions in the given interval

We know that if (\cos x=\frac{1}{2}) and (x\in[0, 2\pi)), then (x = \frac{\pi}{3}, \frac{5\pi}{3}).

Answer:

(\frac{\pi}{3},\frac{5\pi}{3})