Find all solutions of the following equation in the interval $[ 0,2 \pi )$.
$$\csc ^ { 2 } x = \cot x + 1$$
A) $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 4 \pi } { 3 }$
B) $x = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$
C) $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$
D) $x = \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 4 } , \frac { 3 \pi } { 2 }$
E) $x = \frac { \pi } { 4 } , \frac { 4 \pi } { 3 } , \frac { 11 \pi } { 6 }$

Question

Quizplus · Accepted Answer

The given equation is $\csc^2 x = \cot x + 1$. Using the Pythagorean identity, $\csc^2 x = 1 + \cot^2 x$, we can rewrite the equation as $1 + \cot^2 x = \cot x + 1$. Simplifying, we get $\cot^2 x - \cot x = 0$, which factors to $\cot x(\cot x - 1) = 0$. This gives us $\cot x = 0$ or $\cot x = 1$. For $\cot x = 0$, $x = \frac{\pi}{2}, \frac{3\pi}{2}$. For $\cot x = 1$, $x = \frac{\pi}{4}, \frac{5\pi}{4}$. These solutions are within the interval $[0, 2\pi)$.

Find All Solutions of the Following Equation in the Interval [0,2π)[ 0,2 \pi )[0,2π)

Find All Solutions of the Following Equation in the Interval $[ 0,2 \pi )$