Pré-Vestibular ⇒ (UFRGS) Funções

[botelho]

Reduza as seguintes expressões [tex3]\frac{1}{1+sec^{2}\theta } + \frac{1}{1+csc^{2}\theta } + \frac{1}{1+cos^{2}\theta } + \frac{1}{1+sen^{2}\theta }[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\frac{1}{1+sec^{2}\theta } + \frac{1}{1+csc^{2}\theta } + \frac{1}{1+cos^{2}\theta } + \frac{1}{1+sen^{2}\theta }[/tex3]

.
a)1
b)4
c)3
d)2
e)-4

Resposta

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d

[rcompany]

[tex3]
\text{Com }a=\cos \theta, \text{ temos:}\\
\begin{align}
\frac{1}{1+\sec^{2}\theta }+\frac{1}{1+\csc^{2}\theta }+\frac{1}{1+\cos^{2}\theta }+\frac{1}{1+\sen^{2}\theta }&=\frac{1}{1+\frac{1}{a^2}}+\frac{1}{1+\frac{1}{1-a^2}}+\frac{1}{1+a^2}+\frac{1}{2-a^2}\\
&=\frac{a^2}{1+a^2}+\frac{1-a^2}{2-a^2}+\frac{1}{1+a^2}+\frac{1}{2-a^2}\\&=\frac{1+a^2}{1+a^2}+\frac{2-a^2}{2-a^2}\\
&=1+1=2
\end{align}
[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]
\text{Com }a=\cos \theta, \text{ temos:}\\
\begin{align}
\frac{1}{1+\sec^{2}\theta }+\frac{1}{1+\csc^{2}\theta }+\frac{1}{1+\cos^{2}\theta }+\frac{1}{1+\sen^{2}\theta }&=\frac{1}{1+\frac{1}{a^2}}+\frac{1}{1+\frac{1}{1-a^2}}+\frac{1}{1+a^2}+\frac{1}{2-a^2}\\
&=\frac{a^2}{1+a^2}+\frac{1-a^2}{2-a^2}+\frac{1}{1+a^2}+\frac{1}{2-a^2}\\&=\frac{1+a^2}{1+a^2}+\frac{2-a^2}{2-a^2}\\
&=1+1=2
\end{align}
[/tex3]