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Derivation

1. Initial program 0.3

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

2. Initial simplification0.3

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]

3. Taylor expanded around -inf 0.4

   \[\leadsto \color{blue}{\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}}\]
4. Using strategy rm

5. Applied log1p-expm1-u0.4

   \[\leadsto \frac{1 - \frac{{\left(\sin x\right)}^{2}}{\color{blue}{\log_* (1 + (e^{{\left(\cos x\right)}^{2}} - 1)^*)}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}\]

6. Final simplification0.4

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

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))

Details

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