Ejercicios sección 5.7

Ejercicios 1 al 10. Demuestre las identidades trigonométricas propuestas.

  1. sen2  ⁡θ(1+cot2  θ)≡1sen^2\;{⁡\theta}\left(1+cot^2\;{\theta}\right)\equiv 1\\[0.2cm]
  2. tan  ⁡θ+cot  ⁡θ≡sec  ⁡θ⋅csc  ⁡θtan\;{⁡\theta}+cot\;{⁡\theta}\equiv sec\;{⁡\theta}\cdot csc\;{⁡\theta}\\[0.2cm]
  3. tan  ⁡θ+cos  ⁡θ1+sen  ⁡θ≡sec  ⁡θtan\;{⁡\theta}+\cfrac{cos\;{⁡\theta}}{1+sen\;{⁡\theta}}\equiv sec\;{⁡\theta}\\[0.2cm]
  4. 1+sen  ⁡α1−sen  ⁡α−1−sen  ⁡α1+sen  ⁡α≡4⋅tan  ⁡α⋅sec  ⁡α\cfrac{1+sen\;{⁡\alpha}}{1-sen\;{⁡\alpha}}-\cfrac{1-sen\;{⁡\alpha}}{1+sen\;{⁡\alpha}}\equiv 4\cdot tan\;{⁡\alpha}\cdot sec\;{⁡\alpha}\\[0.2cm]
  5. sen  ⁡θ(csc  ⁡θ−sen  θ)≡cos2  ⁡θsen\;{⁡\theta}\left(csc\;{⁡\theta}-sen\;{\theta}\right)\equiv cos^2\;{⁡\theta}\\[0.2cm]
  1. sen2  ⁡θ(1+cot2  θ)≡1sen^2\;{⁡\theta}\left(1+cot^2\;{\theta}\right)\equiv 1\\[0.2cm]
  2. sen  ⁡βcsc  ⁡β+cos  ⁡βsec  ⁡β≡1\cfrac{sen\;{⁡\beta}}{csc\;{⁡\beta}}+\cfrac{cos\;{⁡\beta}}{sec\;{⁡\beta}}\equiv 1\\[0.2cm]
  3. sec  ⁡βtan  ⁡β+cot  ⁡β≡sen  ⁡β\cfrac{sec\;{⁡\beta}}{tan\;{⁡\beta}+cot\;{⁡\beta}}\equiv sen\;{⁡\beta}\\[0.2cm]
  4. tan  ⁡β+cot  ⁡βtan  ⁡β−cot  ⁡β≡sec2  ⁡βtan  ⁡β−1\cfrac{tan\;{⁡\beta}+cot\;{⁡\beta}}{tan\;{⁡\beta}-cot\;{⁡\beta}}\equiv\cfrac{sec^2\;{⁡\beta}}{tan\;{⁡\beta}-1}
  5. 1−2sen2  ⁡θ≡1−tan2  ⁡θ1+tan2  ⁡θ1-2sen^2\;{⁡\theta}\equiv \cfrac{1-tan^2\;{⁡\theta}}{1+tan^2\;{⁡\theta}}\\[0.2cm]