Respuestas sección 5.7
Ejercicios 1 al 10. Demostración de identidades trigonométricas.
$\color{black}\begin{aligned}
sen^2\;{\theta}\left(1+cot^2\;{\theta}\right) &\equiv 1\\[0.3cm]
sen^2\;{\theta}\cdot csc^2\;{\theta} &\equiv \\[0.3cm]
{\color{Red}\bcancel {{\color{black}sen^2\;{\theta}}}}\cdot\cfrac{1}{{\color{Red}\bcancel {{\color{black}sen^2\;{\theta}}}}} &\equiv\\[0.3cm]
1 &\equiv 1
\end{aligned}$
$\hspace{0.1cm}\color{black}\begin{aligned}
tan\;{\theta}+cot\;{\theta} &\equiv sec\;{\theta}\cdot csc\;{\theta}\\[0.3cm]
\cfrac{sen\;{\theta}}{cos\;{\theta}}+\cfrac{cos\;{\theta}}{sen\;{\theta}} &\equiv \\[0.3cm]
\cfrac{sen^2\;{\theta}+cos^2\;{\theta}}{cos\;{\theta}\cdot sen\;{\theta}}&\equiv\\[0.3cm]
\cfrac{1}{cos\;{\theta}\cdot sen\;{\theta}}&\equiv\\[0.3cm]
sec\;{\theta}\cdot csc\;{\theta} &\equiv sec\;{\theta}\cdot csc\;{\theta}
\end{aligned}$
$\color{black}\begin{aligned}
tan\;{\theta}+\cfrac{cos\;{\theta}}{1+sen\;{\theta}} &\equiv sec\;{\theta}\\[0.3cm]
\cfrac{sen\;{\theta}}{cos\;{\theta}}+\cfrac{sen\;{\theta}}{1+sen\;{\theta}}&\equiv \\[0.3cm]
\cfrac{sen\;{\theta}+sen^2\;{\theta}+cos^2\;{\theta}}{cos\;{\theta}\cdot \left(1+sen\;{\theta}\right)}&\equiv\\[0.3cm]
\cfrac{{\color{Red}\bcancel {{\color{black}sen\;{\theta}+1}}}}{cos\;{\theta}\cdot \left({\color{Red}\bcancel {{\color{black}1+sen\;{\theta}}}}\right)}&\equiv\\[0.3cm]
sec\;{\theta} &\equiv sec\;{\theta}
\end{aligned}$
$\color{black}\begin{aligned}
\cfrac{1+sen\;{\alpha}}{1-sen\;{\alpha}}-\cfrac{1-sen\;{\alpha}}{1+sen\;{\alpha}} &\equiv 4\cdot tan\;{\alpha}\cdot sec\;{\alpha}\\[0.3cm]
\cfrac{\left(1+sen\;{\alpha}\right)^2-\left(1-sen\;{\alpha}\right)^2}{1-sen^2\;{\alpha}}&\equiv \\[0.3cm]
\cfrac{{\color{Red}\bcancel {{\color{black}1}}}+2sen\;{\alpha}+{\color{Red}\bcancel {{\color{black}sen^2\;{\alpha}}}}-{\color{Red}\bcancel {{\color{black}1}}}+2sen\;{\alpha}-{\color{Red}\bcancel {{\color{black}sen^2\;{\alpha}}}}}{cos^2\;{\alpha}}&\equiv\\[0.3cm]
\cfrac{4\cdot sen{\alpha}}{cos^2\;{\alpha}}&\equiv\\[0.3cm]
4\cdot\cfrac{ sen{\alpha}}{cos\;{\alpha}}\cdot \cfrac{1}{cos{\alpha}}&\equiv\\[0.3cm]
4\cdot tan{\alpha}\cdot \cfrac{1}{cos{\alpha}} &\equiv \\[0.3cm]
4\cdot tan{\alpha}\cdot sec\;{\alpha} &\equiv 4\cdot tan{\alpha}\cdot sec\;{\alpha}
\end{aligned}$
$\color{black}\begin{aligned}
sen\;{\theta}\left(csc\;{\theta}-sen\;{\theta}\right) &\equiv cos^2\;{\theta}\\[0.3cm]
sen\;{\theta}\cdot\left(\cfrac{1}{sen\;{\theta}}-sen\;{\theta}\right) &\equiv \\[0.3cm]
{\color{Red}\bcancel {{\color{black}sen\;{\theta}}}}\cdot\left(\cfrac{1-sen^2\;{\theta}}{{\color{Red}\bcancel {{\color{black}sen\;{\theta}}}}}\right) &\equiv \\[0.3cm]
cos^2\;{\theta} &\equiv cos^2\;{\theta}
\end{aligned}$
$\color{black}\begin{aligned}
\left(1-cos^2\;{\theta}\right)\cdot\left(1+cot^2\;{\theta}\right) &\equiv 1\\[0.3cm]
sen^2\;{\theta}\cdot csc^2\;{\theta} &\equiv \\[0.3cm]
{\color{Red}\bcancel {{\color{black}sen^2\;{\theta}}}}\cdot\cfrac{1}{{\color{Red}\bcancel {{\color{black}sen^2\;{\theta}}}}} &\equiv\\[0.3cm]
1 &\equiv 1
\end{aligned}$
$\color{black}\begin{aligned}
\cfrac{sen\;{\beta}}{csc\;{\beta}}+\cfrac{cos\;{\beta}}{sec\;{\beta}} &\equiv 1\\[0.3cm]
\cfrac{\cfrac{sen\;{\beta}}{1}}{\cfrac{1}{sen\;{\beta}}}+\cfrac{\cfrac{cos\;{\beta}}{1}}{\cfrac{1}{cos\;{\beta}}} &\equiv \\[0.3cm]
sen^2\;{\beta}+cos^2\;{\beta} &\equiv \\[0.3cm]
1 &\equiv 1
\end{aligned}$
$\color{black}\begin{aligned}
\cfrac{sec\;{\beta}}{tan\;{\beta}+cot\;{\beta}} &\equiv sen\;{\beta}\\[0.3cm]
\cfrac{\cfrac{1}{cos\;{\beta}}}{\cfrac{sen\;{\beta}}{cos\;{\beta}}+\cfrac{cos\;{\beta}}{sen\;{\beta}}} &\equiv \\[0.3cm]
\cfrac{\cfrac{1}{{\color{Red}\bcancel {{\color{black}cos\;{\beta}}}}}}{\cfrac{sen^2\;{\beta}+cos^2\;{\beta}}{sen\;{\beta}\cdot {\color{Red}\bcancel {{\color{black}cos\;{\beta}}}}}} &\equiv \\[0.3cm]
\cfrac{1}{\cfrac{sen^2\;{\beta}+cos^2\;{\beta}}{sen\;{\beta} }} &\equiv \\[0.3cm]
\cfrac{1}{\cfrac{1}{sen\;{\beta} }} &\equiv \\[0.3cm]
sen\;{\beta} &\equiv sen\;{\beta}
\end{aligned}$
$\color{black}\begin{aligned}
\cfrac{tan\;{\beta}+cot\;{\beta}}{tan\;{\beta}-cot\;{\beta}}&\equiv \cfrac{sec^2\;{\beta}}{tan^2\;{\beta}-1}\\[0.4cm]
\cfrac{\cfrac{sen\;{\beta}}{cos\;{\beta}}+\cfrac{cos\;{\beta}}{sen\;{\beta}}}{\cfrac{sen\;{\beta}}{cos\;{\beta}}-\cfrac{cos\;{\beta}}{sen\;{\beta}}} &\equiv \cfrac{\cfrac{1}{cos^2\;{\beta}}}{\cfrac{sen^2\;{\beta}}{cos^2\;{\beta}}-1}\\[1cm]
\cfrac{\cfrac{sen^2\;{\beta}+cos^2\;{\beta}}{sen\;{\beta}\cdot cos\;{\beta}}}{\cfrac{sen^2\;{\beta}-cos^2\;{\beta}}{sen\;{\beta}\cdot cos\;{\beta}}} &\equiv \cfrac{\cfrac{1}{cos^2\;{\beta}}}{\cfrac{sen^2\;{\beta}-cos^2\;{\beta}}{cos^2\;{\beta}}}\\[1cm]
\cfrac{\cfrac{sen^2\;{\beta}+cos^2\;{\beta}}{{\color{Red}\bcancel {{\color{black}sen\;{\beta}\cdot cos\;{\beta}}}}}}{\cfrac{sen^2\;{\beta}-cos^2\;{\beta}}{{\color{Red}\bcancel {{\color{black}sen\;{\beta}\cdot cos\;{\beta}}}}}} &\equiv \cfrac{\cfrac{1}{{\color{Red}\bcancel {{\color{black}cos^2\;{\beta}}}}}}{\cfrac{sen^2\;{\beta}-cos^2\;{\beta}}{{\color{Red}\bcancel {{\color{black}cos^2\;{\beta}}}}}}\\[1cm]
\cfrac{sen^2\;{\beta}+cos^2\;{\beta}}{sen^2\;{\beta}-cos^2\;{\beta}} &\equiv \cfrac{1}{sen^2\;{\beta}-cos^2\;{\beta}}\\[0.4cm]
\color{brown}\cfrac{1}{sen^2\;{\beta}-cos^2\;{\beta}} &\equiv \color{brown}\cfrac{1}{sen^2\;{\beta}-cos^2\;{\beta}}\\[1cm]
\end{aligned}$
$\color{black}\begin{aligned}
1-2sen^2\;{\theta} &\equiv \cfrac{1-tan^2\;{\theta}}{1+tan^2\;{\theta}}\\[0.3cm]
sen^2\;{\theta}+cos^2\;{\theta}-2sen^2\;{\theta} &\equiv \\[0.3cm]
cos^2\;{\theta}-sen^2\;{\theta} &\equiv \\[0.3cm]
\cfrac{cos^2\;{\theta}-sen^2\;{\theta}}{1} &\equiv \\[0.3cm]
\cfrac{cos^2\;{\theta}-sen^2\;{\theta}}{sen^2\;{\theta}+cos^2\;{\theta}} &\equiv \\[0.3cm]
\cfrac{\cfrac{{\color{Red}\bcancel {{\color{black}cos^2\;{\theta}}}}}{{\color{Red}\bcancel {{\color{black}cos^2\;{\theta}}}}}-\cfrac{sen^2\;{\theta}}{cos^2\;{\theta}}}{\cfrac{sen^2\;{\theta}}{cos^2\;{\theta}}+\cfrac{{\color{Red}\bcancel {{\color{black}cos^2\;{\theta}}}}}{{\color{Red}\bcancel {{\color{black}cos^2\;{\theta}}}}}} &\equiv \\[0.3cm]
\cfrac{1-tan^2\;{\theta}}{1+tan^2\;{\theta}} &\equiv \cfrac{1-tan^2\;{\theta}}{1+tan^2\;{\theta}}
\end{aligned}$