Question

Sabendo que tgx – secx = - 21 , determine o valor de tgx + secx.

Answer 1

Observe que, $\text{sec}^2~x=1+\text{tg}^2~x$

Temos que, $\text{tg}~x-\text{sec}~x=-21$.

Assim, $\text{tg}~x=\text{sec}~x-21$.

Elevando os dois lados ao quadrado, obtemos $\text{tg}^2~x=\text{sec}^2~x-42\text{sec}~x+441$.

Por outro lado, $\text{tg}^2~x=\text{sec}^2-1$.

Logo:

$\text{sec}^2~x-1=\text{sec}^2~x-42\text{sec}~x+441$

$42\text{sec}~x=442$

$\text{sec}~x=\dfrac{442}{42}$

$\text{sec}~x=\dfrac{221}{21}$

Assim, $\text{tg}~x=\sqrt{\left(\dfrac{221}{21}\right)^2-1}=\sqrt{\dfrac{48~400}{441}}=\dfrac{220}{21}$.

Portanto, $\text{tg}~x+\text{sec}~x=\dfrac{220}{21}+\dfrac{221}{21}=\dfrac{220+221}{21}=21$.