I have a field of view of 90 with an aspect ratio of 4:3.  The value that comes out of this is 1.0.
If I want to set the aspect ratio to 16:9 with a field of view of 90, I use:
2 * atan((4:3 / 16:9) * tan(1.0 / 2))
Using this I get 1.333307009707560, which makes the image correct to 16:9.
However, if I keep the aspect ratio at 4:3, and lower the FOV to 65, the number that used to be 1.0 is now 0.63707029819488525
So if I want to calculate 16:9 from that, I have to use that number as the source, which gives me 0.849420257806366 for 16:9.

What I want to know is how to figure out why 1.0 lowered to 0.63707.

I'll provide a bunch of FOV values and the number it produces:
I can use a FOV value anywhere between 1 and 160 so if you need more numbers, let me know.
45 -> 0.41421359777450562
65 -> 0.63707029819488525
66 -> 0.64940762519836426
67 -> 0.66188549995422363
68 -> 0.67450851202011108
69 -> 0.68728101253509521
70 -> 0.70020759105682373
71 -> 0.71329307556152344
72 -> 0.72654259204864502
73 -> 0.73996102809906006
74 -> 0.75355404615402222
75 -> 0.76732701063156128
76 -> 0.78128564357757568
77 -> 0.79543590545654297
78 -> 0.80978411436080933
79 -> 0.82433640956878662
80 -> 0.83909958600997925
81 -> 0.85408067703247070
82 -> 0.86928671598434448
83 -> 0.88472527265548706
84 -> 0.90040415525436401
85 -> 0.91633117198944092
86 -> 0.93251502513885498
87 -> 0.94896453619003296
88 -> 0.96568882465362549
89 -> 0.98269736766815186
90 -> 1.00000000000000000