9. -/1 points details my notes larpcalclim4hs 5.3.024. solve the equation. (enter your answers as a comma…

Explanation:

Step1: Rearrange the equation

First, move all terms to one - side: $8\sec(x)\csc(x)-16\csc(x) = 0$. Then factor out $\csc(x)$: $\csc(x)(8\sec(x)-16)=0$.

Step2: Set each factor equal to zero

Case 1: $\csc(x)=0$. But the range of $\csc(x)=\frac{1}{\sin(x)}$ is $(-\infty,-1]\cup[1,\infty)$, so $\csc(x) = 0$ has no solutions. Case 2: $8\sec(x)-16 = 0$. Add 16 to both sides: $8\sec(x)=16$. Then divide by 8: $\sec(x)=2$. Since $\sec(x)=\frac{1}{\cos(x)}$, we have $\frac{1}{\cos(x)} = 2$, so $\cos(x)=\frac{1}{2}$.

Step3: Find the values of x

The general solution for $\cos(x)=\frac{1}{2}$ is $x = 2n\pi\pm\frac{\pi}{3}$, where $n\in\mathbb{Z}$.

Answer:

$x = 2n\pi+\frac{\pi}{3},2n\pi - \frac{\pi}{3},n\in\mathbb{Z}$