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For the a≈ pi *a` /180º part.

That is an identity. That's the degree and radians conversion formula and it applies for any values.

a \(\equiv\) pi *a` /180º

Well     tan a  =  sina / cosa    for small angles cosa ~ 1  thus   tana = ~ sina /1 = sin a

.9 tana =  sina

tana / sina = 1/ .9

1/cosa = 1/.9

.9 = cosa    a =25.84 degrees     Tan 25.84 = .484     sin 25.84 = .4358

Show that for small angles under 20º that tan a ≈ sin a?

Construct a right-angled triangle with angle theta and 3 side lengths = a,b, and c. 

Just like this:

As theta decreases, sin theta=b/c decreases, therefore b decreases, therefore cos theta approaches 1, therefore a ≈ c, therefore b/a ≈ b/c, therefore tan theta ≈ sin theta.

For sin a ≈ a part.....

Use my triangle above.

As theta approaches 0, b approaches 0, b/c approaches 0, therefore sin theta approaches 0

This can be represented as \(\displaystyle\lim_{\theta\rightarrow0}\sin \theta=0\)

Therefore for small values, sin a ≈ a

0.9tan a = sin a

sin a / tan a = 0.9

cos a = 0.9

a = arccos 0.9 ≈ 0.451026811796 rad.