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

Explanation:

Step1: Rewrite tangent function

Recall that $\tan x=\frac{\sin x}{\cos x}$. The given equation $2\cos x + 2\sin x\tan x=4$ becomes $2\cos x+2\sin x\cdot\frac{\sin x}{\cos x}=4$. Multiply through by $\cos x$ (assuming $\cos x\neq0$) to get $2\cos^{2}x + 2\sin^{2}x=4\cos x$.

Step2: Use trigonometric identity

By the Pythagorean - identity $\sin^{2}x+\cos^{2}x = 1$, the left - hand side of the equation $2\cos^{2}x + 2\sin^{2}x=4\cos x$ simplifies to $2(\cos^{2}x+\sin^{2}x)=4\cos x$, so $2 = 4\cos x$.

Step3: Solve for cosine

Divide both sides of the equation $2 = 4\cos x$ by 4 to get $\cos x=\frac{1}{2}$.

Step4: Find solutions in the given interval

In the interval $[0,2\pi)$, the solutions of the equation $\cos x=\frac{1}{2}$ are $x = \frac{\pi}{3}$ and $x=\frac{5\pi}{3}$.

Answer:

$\frac{\pi}{3},\frac{5\pi}{3}$