Question

2. Demostrar las siguientes identidades trigonométricas:

Answer 1

a) $\frac{(2Sen^{2}(\alpha)-1)^{2}}{Sen^{4}(\alpha)-Cos^{4}(\alpha)}=1-2Cos^{2}(\alpha)$

Demostración:

$\frac{(2Sen^{2}(\alpha)-1)^{2}}{Sen^{4}(\alpha)-Cos^{4}(\alpha)}=\frac{(2Sen^{2}(\alpha)-1)^{2}}{(Sen^{2}(\alpha)+Cos^{2}(\alpha))(Sen^{2}(\alpha)-Cos^{2}(\alpha))}\Rightarrow$

$\frac{(2(1-Cos^{2}(\alpha))-1)^{2}}{(1-Cos^{2}(\alpha))-Cos^{2}(\alpha))}=\frac{(1-Cos^{2}(\alpha))^{2}}{1-2Cos^{2}(\alpha)}=1-2Cos^{2}(\alpha)$

b) $\frac{2Tan(\alpha)}{1-Tan^{2}(\alpha)}+\frac{1}{2Cos^{2}(\alpha)-1}=\frac{Cos(\alpha)+Sen(\alpha)}{Cos(\alpha)-Sen(\alpha)}$

Demostración:

$\frac{2Tan(\alpha)}{1-Tan^{2}(\alpha)}+\frac{1}{2Cos^{2}(\alpha)-1}=\frac{2Tan(\alpha)}{1-\frac{Sen^{2}(\alpha)}{Cos^{2}(\alpha)}}+\frac{1}{2Cos^{2}(\alpha)-1}$

$=\frac{Cos^{2}(\alpha)2(\frac{Sen(\alpha)}{Cos(\alpha)})}{Cos^{2}(\alpha)-Sen^{2}(\alpha)}+\frac{1}{2Cos^{2}(\alpha)-1}=\frac{2Sen(\alpha)Cos(\alpha)}{Cos^{2}(\alpha)-Sen^{2}(\alpha)}+\frac{1}{2Cos^{2}(\alpha)-(Cos^{2}(\alpha)+Sen^{2}(\alpha))}$

$=\frac{2Sen(\alpha)Cos(\alpha)+(Cos^{2}(\alpha)+Sen^{2}(\alpha))}{Cos^{2}(\alpha)-Sen^{2}(\alpha)}=\frac{(Sen(\alpha)+Cos(\alpha))^{2}}{Cos^{2}(\alpha)-Sen^{2}(\alpha)}$

$=\frac{(Sen(\alpha)+Cos(\alpha))^{2}}{(Sen(\alpha)+Cos(\alpha))(Cos(\alpha)-Sen(\alpha)}=\frac{Sen(\alpha)+Cos(\alpha)}{Cos(\alpha)-Sen(\alpha)}$

c) $Tan(\alpha)+\frac{1}{Cos^{3}(\alpha)}-\frac{1}{Sec(\alpha)-Tan(\alpha)}=\frac{Sen^{2}(\alpha)}{Cos^{3}(\alpha)}$

Demostración:

$Tan(\alpha)+\frac{1}{Cos^{3}(\alpha)}-\frac{1}{Sec(\alpha)-Tan(\alpha)}=Tan(\alpha)+\frac{1}{Cos^{3}(\alpha)}-\frac{Sec(\alpha)+Tan(\alpha)}{(Sec(\alpha)-Tan(\alpha))(Sec(\alpha)+Tan(\alpha))}$

$Tan(\alpha)+\frac{1}{Cos^{3}(\alpha)}-\frac{Sec(\alpha)+Tan(\alpha)}{Sec^{2}(\alpha)-Tan^{2}(\alpha)}=Tan(\alpha)+\frac{1}{Cos^{3}(\alpha)}-Sec(\alpha)-Tan(\alpha)$

$=\frac{1}{Cos^{3}(\alpha)}-Sec(\alpha)=\frac{1-Sec(\alpha)Cos^{3}(\alpha)}{Cos^{3}(\alpha)}=\frac{1-Cos^{2}(\alpha)}{Cos^{3}(\alpha)}=\frac{Sen^{2}(\alpha)}{Cos^{3}(\alpha)}$

d) $\frac{3Cos^{2}(\alpha)+5Sen(\alpha)-5}{Cos^{2}(\alpha)}=\frac{3Sen(\alpha)-2}{1+Sen(\alpha)}$

Demostración:

$=\frac{3(1-Sen^{2}(\alpha))+5Sen(\alpha)-5}{1-Sen^{2}(\alpha)}=\frac{-3Sen^{2}(\alpha)+5Sen(\alpha)-2}{1-Sen^{2}(\alpha)}=-\frac{3Sen^{2}(\alpha)-5Sen(\alpha)+2}{1-Sen^{2}(\alpha)}$

$=-\frac{(Sen(\alpha)-1)(3Sen(\alpha)-2)}{(1-Sen(\alpha))(1+Sen(\alpha))}=\frac{(Sen(\alpha)-1)(3Sen(\alpha)-2)}{(Sen(\alpha)-1))(1+Sen(\alpha))}=\frac{3Sen(\alpha)-2}{1+Sen(\alpha)}$

e) $\frac{2Sen^{2}(\alpha)+3Cos(\alpha)-3}{Sen^{2}(\alpha)}=\frac{2Cos(\alpha)-1}{1+Cos(\alpha)}$

Demostración:

$\frac{2(1-Cos^{2}(\alpha))+3Cos(\alpha)-3}{1-Cos^{2}(\alpha)}=\frac{-2Cos^{2}(\alpha))+3Cos(\alpha)-1}{1-Cos^{2}(\alpha)}=-\frac{2Cos^{2}(\alpha))-3Cos(\alpha)+1}{1-Cos^{2}(\alpha)}$

$=-\frac{(Cos(\alpha)-1)(2Cos(\alpha)-1)}{(1-Cos(\alpha))(1+Cos(\alpha))}=\frac{(Cos(\alpha)-1)(2Cos(\alpha)-1)}{(Cos(\alpha)-1)(1+Cos(\alpha))}=\frac{2Cos(\alpha)-1}{1+Cos(\alpha)}$