# Struggling to rearrange equation

I'm trying to prove that

$$\dfrac{l}{c+u}+\dfrac{l}{c-u}=\dfrac{2l}{\sqrt{c^2-u^2}}$$

My workings so far:
$$\dfrac{l}{c+u}+\dfrac{l}{c-u},$$
put over common denominator,
$$\dfrac{l(c-u)}{(c-u)(c+u)}+\dfrac{l(c+u)}{(c-u)(c+u)},$$
$$\dfrac{l(c-u)+l(c+u)}{(c-u)(c+u)},$$
expand out,
$$\dfrac{cl-lu+cl+lu}{(c-u)(c+u)},$$
group up and cancel out,
$$\dfrac{2cl}{c^2-u^2},$$

we need to make it equal to
$$\dfrac{2l}{\sqrt{c^2-u^2}}$$
HOW?!?

Last edited:

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vela
Staff Emeritus
Homework Helper
I'm trying to prove that

$$\dfrac{l}{c+u}+\dfrac{l}{c-u}=\dfrac{2l}{\sqrt{c^2-u^2}}$$

My workings so far:
$$\dfrac{l}{c+u}+\dfrac{l}{c-u},$$
put over common denominator,
$$\dfrac{l(c-u)}{(c-u)(c+u)}+\dfrac{l(c+u)}{(c-u)(c+u)},$$
$$\dfrac{l(c-u)+l(c+u)}{(c-u)(c+u)},$$
expand out,
$$\dfrac{cl-lu+cl+lu}{(c-u)(c+u)},$$
group up and cancel out,
$$\dfrac{2cl}{c^2-u^2},$$

we need to make it equal to
$$\dfrac{2l}{\sqrt{c^2-u^2}}$$
HOW?!?
You can't because they're not equal. It's more apparent when you rewrite each slightly:
$$\frac{2cl}{c^2-u^2} = \frac{2cl}{c^2[1-(u/c)^2]} = \frac{2l}{c}\frac{1}{1-(u/c)^2}$$ and
$$\frac{2l}{\sqrt{c^2-u^2}} = \frac{2l}{\sqrt{c^2[1-(u/c)^2]}} = \frac{2l}{c}\frac{1}{\sqrt{1-(u/c)^2}}$$

PeroK