Find the general solution of $3\csc^2 x - 4=0$.

Question

Find the general solution of $3\csc^2 x - 4 =0$.

My Attempt:

$$3\csc^2 x - 4=0$$ $$\csc^2 x =\dfrac {4}{3}$$ $$\csc x = \pm \dfrac {2}{\sqrt {3}}$$

Answer 1

If $\csc x=\pm\frac2{\sqrt3}$, then $\sin x=\pm\frac{\sqrt3}2$, and so $x=\pm\frac{\pi}3+2k\pi$ or $x=\pm\frac{2\pi}3+2k\pi$, for some $k\in\mathbb Z$.

Answer 2

If $\csc^2 x = \frac 43$, then $\cot^2 x = \frac 13$, then $\tan x = \pm\sqrt3$, and so $x = \pm \frac\pi3 + k\pi$, for some $k\in\mathbb Z$.

Answer 3

We can prove $$\sin^2x=\sin^2A\iff\cos^2x=\cos^2A\iff\tan^2x=\tan^2A$$ etc.

Now using Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $,

$$\sin^2x=\sin^2A\iff\sin(x-A)\cdot\sin(x+A)=0\implies x=n\pi\pm A$$ where $n$ is any integer

Here $\csc^2x=\dfrac43\iff\sin^2x=\dfrac34=\sin^2\dfrac\pi3$