1. Interpret the condition

The statement means

so that the expressions involved are well-defined and we exclude those values.

We need values of $\theta \in \left(-\frac\pi2,\frac\pi2\right)$ satisfying both:

$\tan\theta = \cot 5\theta$ and $\sin 2\theta = \cos 4\theta.$

2. Solve $\tan\theta = \cot 5\theta$

Using $\cot x = \tan\left(\frac\pi2 - x\right),$ we get

Hence,

for some integer $k$.

So,

which gives

Now impose $-\frac\pi2 < \theta < \frac\pi2.$

So possible values are:

Also check excluded values $\theta \neq \frac{n\pi}{5}$ for $n=0,\pm1,\pm2$: none of the above are among $0,\pm\frac\pi5,\pm\frac{2\pi}5.$ So all 6 remain.

3. Solve $\sin 2\theta = \cos 4\theta$

Use $\cos 4\theta = \sin\left(\frac\pi2 - 4\theta\right).$ Then

For $\sin A = \sin B$, we have $A=B+2m\pi \quad \text{or} \quad A=\pi-B+2m\pi.$

So either

Case 1:

Case 2:

Within $\left(-\frac\pi2,\frac\pi2\right)$, this gives

and from Case 1,

within the interval. So total distinct solutions of the second equation in the interval are

4. Take common solutions of both equations

From the first equation:

From the second equation:

Their intersection is

Hence the number of values is

5. Compare with stored answer

Stored correct answer = $3$.

Our derived answer also equals $3$, so they agree.