The best way to understand all this I think is to be able to visualize the geometry.

Imagine a circle with radius 1, centered at (0,0).

Now draw a line segment from (0,0) to somewhere on that circle. Note the angle that segment makes with the x-axis.

That point on the circle will have coordinates (x,y).

Let the angle the segment makes with the axis be $\theta$. Then

$\cos(\theta) = x$

$\sin(\theta) = y$

$\tan(\theta) = \dfrac y x$

and this is true for whatever angle/point on the circle you choose.

So for a given $\theta$ there are going to be 2 points on the circle that have a given value for $\cos(\theta)$ and $\sin(\theta)$ and $\tan(\theta)$

Suppose you have a point (x,y) in the first quadrant that makes angle $\theta$ with the x-axis.

From above you can see that the points

(x,y) and (x,-y) will have the same cosine

(x,y) and (-x,y) will have the same sine

(x, y) and (-x,y) will have cosines that are the negative of one another

(x,y) and (x,-y) will have sines that are the negative of one another

so the cosine is preserved when reflecting about the y axis and the sine is preserved when reflecting about the x axis.

the cosine is negated when reflecting about the x axis, and the sine is negated when reflecting about the y axis.

The tangent is thus preserved when reflecting about the origin, i.e. (x,y) -> (-x,-y) and is negated when reflecting about a single axis.

Now with a bit of thought you can turn these reflections into operations on $\theta$

for example a reflection about the y axis is $\theta \to (\pi - \theta)$

see if you can come up with the angular operation for reflection about the x-axis and about the origin.