1. Find set $A$

Given

$$A=\left\{x\in (0,\pi)-\left\{\frac{\pi}{2}\right\}:\log_{2/\pi}|\sin x|+\log_{2/\pi}|\cos x|=2\right\}.$$

Using log property,

$$\log_{2/\pi}(|\sin x|\,|\cos x|)=2.$$

So,

$$|\sin x|\,|\cos x|=\left(\frac{2}{\pi}\right)^2.$$

Since $x\in(0,\pi)$, we have $\sin x>0$, and

$$|\sin x||\cos x|=\sin x\,|\cos x|=\frac{2}{\pi^2}\cdot 2 = \frac{4}{\pi^2}?$$

More directly,

$$|\sin x||\cos x|=\left(\frac{2}{\pi}\right)^2=\frac{4}{\pi^2}.$$

Now use

$$2|\sin x||\cos x|=|\sin 2x|.$$

Hence,

$$|\sin 2x|=2\cdot \frac{4}{\pi^2}=\frac{8}{\pi^2}.$$

Since $0<x<\pi$, we get $0<2x<2\pi$. Let

$$\alpha=\sin^{-1}\left(\frac{8}{\pi^2}\right), \qquad 0<\alpha<\frac{\pi}{2}.$$

Then in $(0,2\pi)$, the equation $|\sin 2x|=\frac{8}{\pi^2}$ has 4 solutions:

$$2x=\alpha,\ \pi-\alpha,\ \pi+\alpha,\ 2\pi-\alpha.$$

Therefore $x$ has 4 corresponding values in $(0,\pi)$, none equal to $\pi/2$. So,

$$n(A)=4.$$

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2. Find set $B$

Given

$$B=\{x\ge 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}.$$

Let

$$t=\sqrt{x}, \qquad t\ge 0.$$

Then the equation becomes

$$t(t-4)-3|t-2|+6=0$$ $$t^2-4t-3|t-2|+6=0.$$

We solve piecewise.

Case 1: $t\ge 2$

Then $|t-2|=t-2$. So,

$$t^2-4t-3(t-2)+6=0$$ $$t^2-7t+12=0$$ $$(t-3)(t-4)=0.$$

Thus,

$$t=3,4.$$

Both satisfy $t\ge 2$. Hence,

$$x=t^2=9,16.$$

Case 2: $0\le t<2$

Then $|t-2|=2-t$. So,

$$t^2-4t-3(2-t)+6=0$$ $$t^2-4t-6+3t+6=0$$ $$t^2-t=0$$ $$t(t-1)=0.$$

Thus,

$$t=0,1.$$

Both satisfy $0\le t<2$. Hence,

$$x=t^2=0,1.$$

Therefore,

$$B=\{0,1,9,16\}, \qquad n(B)=4.$$

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3. Find $A\cup B$

Set $A$ consists of angles in $(0,\pi)$, so its elements are real numbers between $0$ and $\pi$.

Set $B=\{0,1,9,16\}$. Among these, only $1$ lies in $(0,\pi)$. We must check whether $1\in A$.

For $x=1$,

$$|\sin 1||\cos 1|\approx (0.84)(0.54)\approx 0.45,$$

whereas

$$\frac{4}{\pi^2}\approx 0.405.$$

So $1\notin A$. Also $0\notin A$ since $A\subset (0,\pi)$, and $9,16\notin A$. Hence,

$$A\cap B=\varnothing.$$

Therefore,

$$n(A\cup B)=n(A)+n(B)=4+4=8.$$

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4. Compare with stored answer

Derived answer: 8

Stored correct answer: B = 8

So the derived answer agrees with the stored answer.