Verify that √((1-cosx)/(1+cosx)) = (1-cosx)/|sinx|

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 ... $\displaystyle \sqrt{a^2} = |a|$

Thank you, I understand the proof now!

A little clarification in this matter:
|a| = √a^2

is this because its a^2 and not (-a)^2? (So if it were (-a)^2 it'd be -|a| ?)

Thanks!

Quote:

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Originally Posted by SMA777

Thank you, I understand the proof now!

A little clarification in this matter:
|a| = √a^2

is this because its a^2 and not (-a)^2? (So if it were (-a)^2 it'd be -|a| ?)

Thanks!

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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" $\displaystyle a$.