For this problem we're supposed to prove/verify that:
√((1-cosx)/(1+cosx)) = (1-cosx)/|sinx|
Also, if someone could explain why the sinx in the right side has to be absolute value, that would be appreciated. Thank you
For this problem we're supposed to prove/verify that:
√((1-cosx)/(1+cosx)) = (1-cosx)/|sinx|
Also, if someone could explain why the sinx in the right side has to be absolute value, that would be appreciated. Thank you
$\displaystyle \sqrt{\frac{1-\cos{x}}{1+\cos{x}}} \cdot \sqrt{\frac{1-\cos{x}}{1-\cos{x}}} =$
$\displaystyle \frac{1 - \cos{x}}{\sqrt{1 - \cos^2{x}}} =$
$\displaystyle \frac{1 - \cos{x}}{\sqrt{\sin^2{x}}} =$
$\displaystyle \frac{1 - \cos{x}}{|\sin{x}|}$
basic algebra ... $\displaystyle \sqrt{a^2} = |a|$
There are always two square roots to any number.
i.e. $\displaystyle \sqrt{a^2} = a$ or $\displaystyle \sqrt{a^2} = -a$
Notice that both answers have a "size" of $\displaystyle a$.
This "size" is denoted with an absolute value.
So $\displaystyle \sqrt{a^2} = |a|$ is correct, because it's saying the answer is something of "size" $\displaystyle a$.