\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

↓

\[\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}\]

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Derivation

Initial program 0.3
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
Initial simplification0.3
\[\leadsto \frac{1 - \tan x \cdot \tan x}{\tan x \cdot \tan x + 1}\]
Taylor expanded around inf 0.4
\[\leadsto \color{blue}{\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}}\]
Final simplification0.4
\[\leadsto \frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}\]

herbie shell --seed 2018346 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))