Estude! Com Prof. Caju

(UECE) Se p=[tex3]\frac{sen 40°}{sen20°} = \frac{cos40°}{cos20°}[/tex3] , então [tex3]p^{2}[/tex3]-1 é igual a
a)[tex3]sen^2[/tex3] 20°
b)[tex3]cos^2[/tex3] 20°
c)[tex3]tg^2[/tex3] 20°
d)[tex3]cotg^2[/tex3] 20°

Observe que:
[tex3]sen40=2.sen20.cos20[/tex3]
[tex3]cos40=2.cos^2 20 -1[/tex3]

Substituindo estes valores, temos:
[tex3]p=\frac{sen40}{sen20}=\frac{cos40}{cos20}[/tex3]

[tex3]p= \frac{2.sen20.cos20}{sen20}-\frac{2.cos^2 20 -1}{cos20}[/tex3]

[tex3]p= 2.cos20- 2.cos20 + \frac{1}{cos20}[/tex3]

[tex3]p=\frac{1}{cos20}[/tex3]

logo,
[tex3]p^2=\frac{1}{cos^2 20}[/tex3]

[tex3]p^2-1=\frac{1}{cos^2 20}-1[/tex3]

[tex3]p^2-1=\frac{1-cos^2 20}{cos^2 20}[/tex3]

[tex3]p^2-1=\frac{(sen^2 20 +cos^2 20)-cos^2 20}{cos^2 20}[/tex3]

[tex3]p^2-1=\frac{sen^2 20 +\cancel{cos^2 20}-\cancel{cos^2 20}}{cos^2 20}[/tex3]

[tex3]p^2-1=\frac{sen^2 20}{cos^2 20}=tan^2 20[/tex3]

Portanto,
[tex3]\boxed{p^2-1=tan^2 20}[/tex3]

Abraço.