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

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) $x = \frac { \pi } { 4 } , \frac { 4 \pi } { 3 } , \frac { 11 \pi } { 6 }$

Quizplus · Accepted Answer

We start by using the trigonometric identities $\csc^2 x=\frac{1}{\sin^2 x}$ and $\cot x = \frac{\cos x}{\sin x}$ to rewrite the equation as
$$\frac{1}{\sin^2 x} = \frac{\cos x}{\sin x} + 1.$$
Multiplying both sides by $\sin^2 x$ and rearranging, we get
$$\cos x = \sin^3 x - \sin^2 x.$$
Now, we use the identity $\cos^2 x = 1 - \sin^2 x$ to eliminate $\cos x$:
$$(1 - \sin^2 x) = \sin^3 x - \sin^2 x.$$
Simplifying and rearranging, we get
$$\sin^3 x - \sin^2 x - \sin x + 1 = 0.$$
The left-hand side factors as $(\sin x - 1)(\sin^2 x - 1)$. Therefore, the solutions to the equation are given by the solutions to the equations $\sin x = 1$ and $\sin x = \pm 1$, which are $x = \frac{\pi}{2}$ and $x = 0,\pi$, respectively. Finally, we note that $\frac{\pi}{2}$ is not in the given interval, so the solutions in the interval are $x=0$ and $x=\pi$. However, we need to check if these values satisfy the original equation:
$$\csc^2 0 = \cot 0 + 1 \quad \Rightarrow \quad 1 = \infty,$$
and
$$\csc^2 \pi = \cot \pi + 1 \quad \Rightarrow \quad 1 = -1.$$
Therefore, the only solutions in the given interval are $x = \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$, which are both in choice D.

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 $[ 0,2 \pi )$