Respuestas sección 5.7

Ejercicios 1 al 10. Demostración de identidades trigonométricas.

  1. sen2  ⁡θ(1+cot2  θ)≡1sen2  ⁡θ⋅csc2  θ≡sen2  ⁡θ⋅1sen2  θ≡1≡1\color{black}\begin{aligned} sen^2\;{⁡\theta}\left(1+cot^2\;{\theta}\right) &\equiv 1\\[0.3cm] sen^2\;{⁡\theta}\cdot csc^2\;{\theta} &\equiv \\[0.3cm] {\color{Red}\bcancel {{\color{black}sen^2\;{⁡\theta}}}}\cdot\cfrac{1}{{\color{Red}\bcancel {{\color{black}sen^2\;{\theta}}}}} &\equiv\\[0.3cm] 1 &\equiv 1 \end{aligned}
  2. tan  ⁡θ+cot  ⁡θ≡sec  ⁡θ⋅csc  ⁡θsen  ⁡θcos  ⁡θ+cos  ⁡θsen  ⁡θ≡sen2  ⁡θ+cos2  ⁡θcos  ⁡θ⋅sen  ⁡θ≡1cos  ⁡θ⋅sen  ⁡θ≡sec  ⁡θ⋅csc  ⁡θ≡sec  ⁡θ⋅csc  ⁡θ\hspace{0.1cm}\color{black}\begin{aligned} tan\;{⁡\theta}+cot\;{⁡\theta} &\equiv sec\;{⁡\theta}\cdot csc\;{⁡\theta}\\[0.3cm] \cfrac{sen\;{⁡\theta}}{cos\;{⁡\theta}}+\cfrac{cos\;{⁡\theta}}{sen\;{⁡\theta}} &\equiv \\[0.3cm] \cfrac{sen^2\;{⁡\theta}+cos^2\;{⁡\theta}}{cos\;{⁡\theta}\cdot sen\;{⁡\theta}}&\equiv\\[0.3cm] \cfrac{1}{cos\;{⁡\theta}\cdot sen\;{⁡\theta}}&\equiv\\[0.3cm] sec\;{⁡\theta}\cdot csc\;{⁡\theta} &\equiv sec\;{⁡\theta}\cdot csc\;{⁡\theta} \end{aligned}
  3. tan  ⁡θ+cos  ⁡θ1+sen  ⁡θ≡sec  ⁡θsen  ⁡θcos  ⁡θ+sen  ⁡θ1+sen  ⁡θ≡sen  ⁡θ+sen2  ⁡θ+cos2  ⁡θcos  ⁡θ⋅(1+sen  ⁡θ)≡sen  ⁡θ+1cos  ⁡θ⋅(1+sen  ⁡θ)≡sec  ⁡θ≡sec  ⁡θ\color{black}\begin{aligned} tan\;{⁡\theta}+\cfrac{cos\;{⁡\theta}}{1+sen\;{⁡\theta}} &\equiv sec\;{⁡\theta}\\[0.3cm] \cfrac{sen\;{⁡\theta}}{cos\;{⁡\theta}}+\cfrac{sen\;{⁡\theta}}{1+sen\;{⁡\theta}}&\equiv \\[0.3cm] \cfrac{sen\;{⁡\theta}+sen^2\;{⁡\theta}+cos^2\;{⁡\theta}}{cos\;{⁡\theta}\cdot \left(1+sen\;{⁡\theta}\right)}&\equiv\\[0.3cm] \cfrac{{\color{Red}\bcancel {{\color{black}sen\;{⁡\theta}+1}}}}{cos\;{⁡\theta}\cdot \left({\color{Red}\bcancel {{\color{black}1+sen\;{⁡\theta}}}}\right)}&\equiv\\[0.3cm] sec\;{⁡\theta} &\equiv sec\;{⁡\theta} \end{aligned}
  4. 1+sen  ⁡α1−sen  ⁡α−1−sen  ⁡α1+sen  ⁡α≡4⋅tan  ⁡α⋅sec  ⁡α(1+sen  ⁡α)2−(1−sen  ⁡α)21−sen2  ⁡α≡1+2sen  ⁡α+sen2  ⁡α−1+2sen  ⁡α−sen2  ⁡αcos2  ⁡α≡4⋅senαcos2  ⁡α≡4⋅senαcos  ⁡α⋅1cosα≡4⋅tanα⋅1cosα≡4⋅tanα⋅sec  ⁡α≡4⋅tanα⋅sec  ⁡α\color{black}\begin{aligned} \cfrac{1+sen\;{⁡\alpha}}{1-sen\;{⁡\alpha}}-\cfrac{1-sen\;{⁡\alpha}}{1+sen\;{⁡\alpha}} &\equiv 4\cdot tan\;{⁡\alpha}\cdot sec\;{⁡\alpha}\\[0.3cm] \cfrac{\left(1+sen\;{⁡\alpha}\right)^2-\left(1-sen\;{⁡\alpha}\right)^2}{1-sen^2\;{⁡\alpha}}&\equiv \\[0.3cm] \cfrac{{\color{Red}\bcancel {{\color{black}1}}}+2sen\;{⁡\alpha}+{\color{Red}\bcancel {{\color{black}sen^2\;{⁡\alpha}}}}-{\color{Red}\bcancel {{\color{black}1}}}+2sen\;{⁡\alpha}-{\color{Red}\bcancel {{\color{black}sen^2\;{⁡\alpha}}}}}{cos^2\;{⁡\alpha}}&\equiv\\[0.3cm] \cfrac{4\cdot sen{\alpha}}{cos^2\;{⁡\alpha}}&\equiv\\[0.3cm] 4\cdot\cfrac{ sen{\alpha}}{cos\;{⁡\alpha}}\cdot \cfrac{1}{cos{\alpha}}&\equiv\\[0.3cm] 4\cdot tan{\alpha}\cdot \cfrac{1}{cos{\alpha}} &\equiv \\[0.3cm] 4\cdot tan{\alpha}\cdot sec\;{⁡\alpha} &\equiv 4\cdot tan{\alpha}\cdot sec\;{⁡\alpha} \end{aligned}
  5. sen  ⁡θ(csc  ⁡θ−sen  θ)≡cos2  ⁡θsen  ⁡θ⋅(1sen  ⁡θ−sen  ⁡θ)≡sen  ⁡θ⋅(1−sen2  ⁡θsen  ⁡θ)≡cos2  ⁡θ≡cos2  ⁡θ\color{black}\begin{aligned} sen\;{⁡\theta}\left(csc\;{⁡\theta}-sen\;{\theta}\right) &\equiv cos^2\;{⁡\theta}\\[0.3cm] sen\;{⁡\theta}\cdot\left(\cfrac{1}{sen\;{⁡\theta}}-sen\;{⁡\theta}\right) &\equiv \\[0.3cm] {\color{Red}\bcancel {{\color{black}sen\;{⁡\theta}}}}\cdot\left(\cfrac{1-sen^2\;{⁡\theta}}{{\color{Red}\bcancel {{\color{black}sen\;{⁡\theta}}}}}\right) &\equiv \\[0.3cm] cos^2\;{⁡\theta} &\equiv cos^2\;{⁡\theta} \end{aligned}
  6. (1−cos2  ⁡θ)⋅(1+cot2  θ)≡1sen2  ⁡θ⋅csc2  θ≡sen2  ⁡θ⋅1sen2  θ≡1≡1\color{black}\begin{aligned} \left(1-cos^2\;{⁡\theta}\right)\cdot\left(1+cot^2\;{\theta}\right) &\equiv 1\\[0.3cm] sen^2\;{⁡\theta}\cdot csc^2\;{\theta} &\equiv \\[0.3cm] {\color{Red}\bcancel {{\color{black}sen^2\;{⁡\theta}}}}\cdot\cfrac{1}{{\color{Red}\bcancel {{\color{black}sen^2\;{\theta}}}}} &\equiv\\[0.3cm] 1 &\equiv 1 \end{aligned}
  7. sen  ⁡βcsc  ⁡β+cos  ⁡βsec  ⁡β≡1sen  ⁡β11sen  ⁡β+cos  ⁡β11cos  ⁡β≡sen2  ⁡β+cos2  ⁡β≡1≡1\color{black}\begin{aligned} \cfrac{sen\;{⁡\beta}}{csc\;{⁡\beta}}+\cfrac{cos\;{⁡\beta}}{sec\;{⁡\beta}} &\equiv 1\\[0.3cm] \cfrac{\cfrac{sen\;{⁡\beta}}{1}}{\cfrac{1}{sen\;{⁡\beta}}}+\cfrac{\cfrac{cos\;{⁡\beta}}{1}}{\cfrac{1}{cos\;{⁡\beta}}} &\equiv \\[0.3cm] sen^2\;{⁡\beta}+cos^2\;{⁡\beta} &\equiv \\[0.3cm] 1 &\equiv 1 \end{aligned}
  1. sec  ⁡βtan  ⁡β+cot  ⁡β≡sen  ⁡β1cos  ⁡βsen  ⁡βcos  ⁡β+cos  ⁡βsen  ⁡β≡1cos  ⁡βsen2  ⁡β+cos2  ⁡βsen  ⁡β⋅cos  ⁡β≡1sen2  ⁡β+cos2  ⁡βsen  ⁡β≡11sen  ⁡β≡sen  ⁡β≡sen  ⁡β\color{black}\begin{aligned} \cfrac{sec\;{⁡\beta}}{tan\;{⁡\beta}+cot\;{⁡\beta}} &\equiv sen\;{⁡\beta}\\[0.3cm] \cfrac{\cfrac{1}{cos\;{⁡\beta}}}{\cfrac{sen\;{⁡\beta}}{cos\;{⁡\beta}}+\cfrac{cos\;{⁡\beta}}{sen\;{⁡\beta}}} &\equiv \\[0.3cm] \cfrac{\cfrac{1}{{\color{Red}\bcancel {{\color{black}cos\;{⁡\beta}}}}}}{\cfrac{sen^2\;{⁡\beta}+cos^2\;{⁡\beta}}{sen\;{⁡\beta}\cdot {\color{Red}\bcancel {{\color{black}cos\;{⁡\beta}}}}}} &\equiv \\[0.3cm] \cfrac{1}{\cfrac{sen^2\;{⁡\beta}+cos^2\;{⁡\beta}}{sen\;{⁡\beta} }} &\equiv \\[0.3cm] \cfrac{1}{\cfrac{1}{sen\;{⁡\beta} }} &\equiv \\[0.3cm] sen\;{⁡\beta} &\equiv sen\;{⁡\beta} \end{aligned}
  2. tan  ⁡β+cot  ⁡βtan  ⁡β−cot  ⁡β≡sec2  ⁡βtan2  ⁡β−1sen  ⁡βcos  ⁡β+cos  ⁡βsen  ⁡βsen  ⁡βcos  ⁡β−cos  ⁡βsen  ⁡β≡1cos2  ⁡βsen2  ⁡βcos2  ⁡β−1sen2  ⁡β+cos2  ⁡βsen  ⁡β⋅cos  ⁡βsen2  ⁡β−cos2  ⁡βsen  ⁡β⋅cos  ⁡β≡1cos2  ⁡βsen2  ⁡β−cos2  ⁡βcos2  ⁡βsen2  ⁡β+cos2  ⁡βsen  ⁡β⋅cos  ⁡βsen2  ⁡β−cos2  ⁡βsen  ⁡β⋅cos  ⁡β≡1cos2  ⁡βsen2  ⁡β−cos2  ⁡βcos2  ⁡βsen2  ⁡β+cos2  ⁡βsen2  ⁡β−cos2  ⁡β≡1sen2  ⁡β−cos2  ⁡β1sen2  ⁡β−cos2  ⁡β≡1sen2  ⁡β−cos2  ⁡β\color{black}\begin{aligned} \cfrac{tan\;{⁡\beta}+cot\;{⁡\beta}}{tan\;{⁡\beta}-cot\;{⁡\beta}}&\equiv \cfrac{sec^2\;{⁡\beta}}{tan^2\;{⁡\beta}-1}\\[0.4cm] \cfrac{\cfrac{sen\;{⁡\beta}}{cos\;{⁡\beta}}+\cfrac{cos\;{⁡\beta}}{sen\;{⁡\beta}}}{\cfrac{sen\;{⁡\beta}}{cos\;{⁡\beta}}-\cfrac{cos\;{⁡\beta}}{sen\;{⁡\beta}}} &\equiv \cfrac{\cfrac{1}{cos^2\;{⁡\beta}}}{\cfrac{sen^2\;{⁡\beta}}{cos^2\;{⁡\beta}}-1}\\[1cm] \cfrac{\cfrac{sen^2\;{⁡\beta}+cos^2\;{⁡\beta}}{sen\;{⁡\beta}\cdot cos\;{⁡\beta}}}{\cfrac{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}}{sen\;{⁡\beta}\cdot cos\;{⁡\beta}}} &\equiv \cfrac{\cfrac{1}{cos^2\;{⁡\beta}}}{\cfrac{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}}{cos^2\;{⁡\beta}}}\\[1cm] \cfrac{\cfrac{sen^2\;{⁡\beta}+cos^2\;{⁡\beta}}{{\color{Red}\bcancel {{\color{black}sen\;{⁡\beta}\cdot cos\;{⁡\beta}}}}}}{\cfrac{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}}{{\color{Red}\bcancel {{\color{black}sen\;{⁡\beta}\cdot cos\;{⁡\beta}}}}}} &\equiv \cfrac{\cfrac{1}{{\color{Red}\bcancel {{\color{black}cos^2\;{⁡\beta}}}}}}{\cfrac{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}}{{\color{Red}\bcancel {{\color{black}cos^2\;{⁡\beta}}}}}}\\[1cm] \cfrac{sen^2\;{⁡\beta}+cos^2\;{⁡\beta}}{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}} &\equiv \cfrac{1}{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}}\\[0.4cm] \color{brown}\cfrac{1}{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}} &\equiv \color{brown}\cfrac{1}{sen^2\;{⁡\beta}-cos^2\;{⁡\beta}}\\[1cm] \end{aligned}
  3. 1−2sen2  ⁡θ≡1−tan2  ⁡θ1+tan2  ⁡θsen2  ⁡θ+cos2  ⁡θ−2sen2  ⁡θ≡cos2  ⁡θ−sen2  ⁡θ≡cos2  ⁡θ−sen2  ⁡θ1≡cos2  ⁡θ−sen2  ⁡θsen2  ⁡θ+cos2  ⁡θ≡cos2  ⁡θcos2  ⁡θ−sen2  ⁡θcos2  ⁡θsen2  ⁡θcos2  ⁡θ+cos2  ⁡θcos2  ⁡θ≡1−tan2  ⁡θ1+tan2  ⁡θ≡1−tan2  ⁡θ1+tan2  ⁡θ\color{black}\begin{aligned} 1-2sen^2\;{⁡\theta} &\equiv \cfrac{1-tan^2\;{⁡\theta}}{1+tan^2\;{⁡\theta}}\\[0.3cm] sen^2\;{⁡\theta}+cos^2\;{⁡\theta}-2sen^2\;{⁡\theta} &\equiv \\[0.3cm] cos^2\;{⁡\theta}-sen^2\;{⁡\theta} &\equiv \\[0.3cm] \cfrac{cos^2\;{⁡\theta}-sen^2\;{⁡\theta}}{1} &\equiv \\[0.3cm] \cfrac{cos^2\;{⁡\theta}-sen^2\;{⁡\theta}}{sen^2\;{⁡\theta}+cos^2\;{⁡\theta}} &\equiv \\[0.3cm] \cfrac{\cfrac{{\color{Red}\bcancel {{\color{black}cos^2\;{⁡\theta}}}}}{{\color{Red}\bcancel {{\color{black}cos^2\;{⁡\theta}}}}}-\cfrac{sen^2\;{⁡\theta}}{cos^2\;{⁡\theta}}}{\cfrac{sen^2\;{⁡\theta}}{cos^2\;{⁡\theta}}+\cfrac{{\color{Red}\bcancel {{\color{black}cos^2\;{⁡\theta}}}}}{{\color{Red}\bcancel {{\color{black}cos^2\;{⁡\theta}}}}}} &\equiv \\[0.3cm] \cfrac{1-tan^2\;{⁡\theta}}{1+tan^2\;{⁡\theta}} &\equiv \cfrac{1-tan^2\;{⁡\theta}}{1+tan^2\;{⁡\theta}} \end{aligned}