How to determine a general arithmetic sequence formula for two intersecting trig function

Question

I have equations out of two trigonometric functions.
For example

• $\cos(4\alpha$) = -$\sin(5\alpha)$
• $\tan(0.5\alpha$) = 2 $\sin(\alpha)$

How can I determine a general arithmetic sequence formula which gives me n-th positive angle where the two trig functions intersects?

By drawing the graph I found out the intersecting points..

1. 30, 70, 110... --> 30° + n * 40
2. 0, 120, 240... --> n * 120

Whereby n starts from 0.

Is there a better way to find these arithemetic sequence formula without drawing the graph?

Answer 1

$$\cos4\alpha=-\sin5\alpha=\cos\left(90^\circ+5\alpha\right)$$

$$\iff4\alpha=360^\circ m\pm\left(90^\circ+5\alpha\right)$$ where $m$ is any integer

Consider '+', '-' sign one by one.

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For the second one use $$\sin2y=\frac{2\tan y}{1+\tan^2y}$$