Ejercicios sección 5.7

Ejercicios 1 al 10. Demuestre las identidades trigonométricas propuestas.

1. $sen^2\;{⁡\theta}\left(1+cot^2\;{\theta}\right)\equiv 1\\[0.2cm]$
2. $tan\;{⁡\theta}+cot\;{⁡\theta}\equiv sec\;{⁡\theta}\cdot csc\;{⁡\theta}\\[0.2cm]$
3. $tan\;{⁡\theta}+\cfrac{cos\;{⁡\theta}}{1+sen\;{⁡\theta}}\equiv sec\;{⁡\theta}\\[0.2cm]$
4. $\cfrac{1+sen\;{⁡\alpha}}{1-sen\;{⁡\alpha}}-\cfrac{1-sen\;{⁡\alpha}}{1+sen\;{⁡\alpha}}\equiv 4\cdot tan\;{⁡\alpha}\cdot sec\;{⁡\alpha}\\[0.2cm]$
5. $sen\;{⁡\theta}\left(csc\;{⁡\theta}-sen\;{\theta}\right)\equiv cos^2\;{⁡\theta}\\[0.2cm]$

6. $sen^2\;{⁡\theta}\left(1+cot^2\;{\theta}\right)\equiv 1\\[0.2cm]$
7. $\cfrac{sen\;{⁡\beta}}{csc\;{⁡\beta}}+\cfrac{cos\;{⁡\beta}}{sec\;{⁡\beta}}\equiv 1\\[0.2cm]$
8. $\cfrac{sec\;{⁡\beta}}{tan\;{⁡\beta}+cot\;{⁡\beta}}\equiv sen\;{⁡\beta}\\[0.2cm]$
9. $\cfrac{tan\;{⁡\beta}+cot\;{⁡\beta}}{tan\;{⁡\beta}-cot\;{⁡\beta}}\equiv\cfrac{sec^2\;{⁡\beta}}{tan\;{⁡\beta}-1}$
10. $1-2sen^2\;{⁡\theta}\equiv \cfrac{1-tan^2\;{⁡\theta}}{1+tan^2\;{⁡\theta}}\\[0.2cm]$