Answer
The solution set is $$\{1.3181+2n\pi,4.9651+2n\pi, n\in Z\}$$
Work Step by Step
$$3\csc^2\frac{x}{2}=2\sec x$$
Here we encounter both functions for angle $x$ and half-angle $\frac{x}{2}$. Currently there is no available identity to deal with half-angle for secant and cosecant functions. So we need to switch them back to sine and cosine functions.
- Recall the identity: $\csc\frac{x}{2}=\frac{1}{\sin\frac{x}{2}}$ and $\sec x=\frac{1}{\cos x}$
$$\frac{3}{\sin^2\frac{x}{2}}=\frac{2}{\cos x}$$
$$3\cos x=2\sin^2\frac{x}{2}$$
- Recall the identity: $\cos2x=1-2\sin^2x$
Thus, we can deduce that $$\cos x=1-2\sin^2\frac{x}{2}$$
$$2\sin^2\frac{x}{2}=1-\cos x$$
Apply back to the equation:
$$3\cos x=1-\cos x$$
$$4\cos x=1$$
$$\cos x=\frac{1}{4}$$
1) First, we solve the equation over the interval $[0,2\pi)$
For $\cos x=\frac{1}{4}$, we have
$$x=\cos^{-1}\frac{1}{4}$$
$$x\approx1.3181$$
Also, over the interval, there is one more value of $x$ where $\cos x=\frac{1}{4}$, which is $x\approx2\pi-1.3181\approx4.9651$
Therefore, $$x=\{1.3181,4.9651\}$$
2) Solve the equation for all solutions
Cosine function has period $2\pi$, so we would add $2\pi$ to all solutions found in part 1) for $x$.
$$x=\{1.3181+2n\pi,4.9651+2n\pi, n\in Z\}$$
