Question

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

Answer 1

@LolWolf, @AccessDenied, @lgbasallote

Answer 2

wait, thats wrong

Answer 3

sinh(3t)/sinh(t)=1+2cosh(2t) for t≠0

Answer 4

they aren't equal...try again

Answer 5

ok finally

Answer 6

@Algebraic! do you know how to do it?

Answer 7

yep

Answer 8

\[ \text{sinh} \; x = \frac{e^x - e^{-x}}{2} \]
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.

Answer 9

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.

Answer 10

@AccessDenied YOU ROCK, thanks so much!!

Answer 11

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

Answer 12

and, I'm glad to help! :)

Answer 13

thanks so much again

Answer 14

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

Answer 15

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

Answer 16

lol, thanks for you valiant effort @LolWolf

Answer 17

Valiantly late, haha, but, yes