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Derivation

1. Initial program 0.3

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
2. Using strategy rm

3. Applied div-sub0.4

   \[\leadsto \color{blue}{\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}}\]
4. Using strategy rm

5. Applied tan-quot0.4

   \[\leadsto \frac{1}{1 + \tan x \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

6. Applied associate-*r/0.4

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
7. Using strategy rm

8. Applied frac-sub0.4

   \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + \tan x \cdot \tan x\right) - \left(1 + \frac{\tan x \cdot \sin x}{\cos x}\right) \cdot \left(\tan x \cdot \tan x\right)}{\left(1 + \frac{\tan x \cdot \sin x}{\cos x}\right) \cdot \left(1 + \tan x \cdot \tan x\right)}}\]

9. Simplified0.5

   \[\leadsto \frac{\color{blue}{1 - \frac{\sin x}{\cos x} \cdot {\left(\tan x\right)}^{3}}}{\left(1 + \frac{\tan x \cdot \sin x}{\cos x}\right) \cdot \left(1 + \tan x \cdot \tan x\right)}\]

10. Final simplification0.5

    \[\leadsto \frac{1 - {\left(\tan x\right)}^{3} \cdot \frac{\sin x}{\cos x}}{\left(1 + \frac{\sin x \cdot \tan x}{\cos x}\right) \cdot \left(\tan x \cdot \tan x + 1\right)}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))

Details

Time bar (total: 25.8s)Debug log