The solution set of the equation $\cot ^ { 2 } x - 1 = \csc x$ on $[ 0,2 \pi ]$ is

A) $\left\{ \frac { \pi } { 6 } \right\}$
B) $\left\{ \frac { 3 \pi } { 2 } \right\}$
C) $\left\{ \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \right\}$
D) $\left\{ \frac { 3 \pi } { 2 } , \frac { 5 \pi } { 6 } \right\}$
E) $\left\{ \frac { \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 5 \pi } { 6 } \right\}$

Question

The solution set of the equation $\cot ^ { 2 } x - 1 = \csc x$ on $[ 0,2 \pi ]$ is

A) $\left\{ \frac { \pi } { 6 } \right\}$
B) $\left\{ \frac { 3 \pi } { 2 } \right\}$
C) $\left\{ \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \right\}$
D) $\left\{ \frac { 3 \pi } { 2 } , \frac { 5 \pi } { 6 } \right\}$
E) $\left\{ \frac { \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 5 \pi } { 6 } \right\}$

Quizplus · Accepted Answer

Using trigonometric identities, $\cot^2x - 1 = \csc x$ can be rewritten as $\csc^2x - 1 = \csc x$ (since $\cot^2x + 1 = \csc^2x$). This simplifies to $\csc^2x - \csc x - 1 = 0$. Solving this quadratic equation in terms of $\csc x$, we find the solutions that correspond to the angles $\frac{\pi}{6}$, $\frac{5\pi}{6}$, and $\frac{3\pi}{2}$ within the given interval $[0, 2\pi]$.

The Solution Set of the Equation cot⁡2x−1=csc⁡x\cot ^ { 2 } x - 1 = \csc xcot2x−1=cscx

The Solution Set of the Equation $\cot ^ { 2 } x - 1 = \csc x$