## baldymcgee6 3 years ago Show that sinh(3t)*sinh(t)=1+2cosh(2t) for t≠0

1. baldymcgee6

@LolWolf, @AccessDenied, @lgbasallote

2. baldymcgee6

wait, thats wrong

3. baldymcgee6

sinh(3t)/sinh(t)=1+2cosh(2t) for t≠0

4. Algebraic!

they aren't equal...try again

5. Algebraic!

ok finally

6. baldymcgee6

@Algebraic! do you know how to do it?

7. Algebraic!

yep

8. AccessDenied

$\text{sinh} \; x = \frac{e^x - e^{-x}}{2}$ We can rewrite the left-hand side to match the right-hand side. $\frac{e^{3x} - e^{-3x}}{2} \div \frac{e^x - e^{-x}}{2} \\ \frac{e^{3x} - e^{-3x}}{\cancel{2}} \times \frac{\cancel{2}}{e^x - e^{-x}}\\ \frac{e^{3x} - e^{-3x}}{e^x - e^{-x}} \\ \frac{(e^x)^3 - (e^{-x})^3}{e^x - e^{-x}}$ If we consider the numerator as a difference of cubes, we can factor it like this: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$. Notice that this creates a factor in the denominator that is also in the numerator.

9. AccessDenied

When the factors cancel, this remains: $\frac{\cancel{(e^x - e^{-x})}((e^x)^2 + e^x e^{-x} + (e^{-x})^2)}{\cancel{e^x - e^{-x}}} \\ = e^{2x} + 1 + e^{-2x}$ Which starts to look a lot like 2cosh 2x + 1, it should be simple manipulation to justify that from there.

10. baldymcgee6

@AccessDenied YOU ROCK, thanks so much!!

11. AccessDenied

I should note that I am using x instead of t. My bad. :P

12. AccessDenied

and, I'm glad to help! :)

13. baldymcgee6

thanks so much again

14. Algebraic!

don't forget, you can redeem your medals for cash prizes at the end of every month.

15. LolWolf

So we know: $\sinh x=\frac{e^x-e^{-x}}{2}$Therefore: $\sinh 3x=\frac{e^{3x}-e^{-3x}}{2}$So: $\frac{\sinh(3t)}{\sinh(t)}=\frac{2}{e^x-e^{-x}}\cdot\frac{e^{3x}-e^{-3x}}{2}=\frac{2e^x}{e^{2x}-1}\cdot\frac{e^{6x}-1}{2e^{3x}}=\\ \frac{2e^x}{e^{2x}-1}\cdot\frac{e^{6x}-1}{2e^{3x}}=\frac{(e^{2x}-1)(e^{4x}+e^{2x}+1)}{e^{2x}(e^{2x}-1)}=\\ \frac{e^{4x}+e^{2x}+1}{e^{2x}}=e^{2x}+1+e^{-2x}=1+2\cosh x$Ahh, this takes forever... and I mad a mistake halfway through, so I had to restart... anyways, +1 internets to @AccessDenied

16. baldymcgee6

lol, thanks for you valiant effort @LolWolf

17. LolWolf

Valiantly late, haha, but, yes