Solving Exponential Form Homework: sinh(3x)=3sinh(x)+4sinh^3(x)

[Pietair]

Homework Statement

Prove that:
$$sinh(3x)=3sinh(x)+4sinh^{3}(x)$$

2. The attempt at a solution
I know that:
$$sinh(3x)=0.5(e^{3x}-e^{-3x})$$

and:
$$3sinh(x)=1.5(e^{x}-e^{-x})$$

But I have no idea how to rewrite $$4sinh^{3}(x)$$ in exponential form...

 

[phsopher]

$$sinh^{3}(x) = [0.5(e^{x}-e^{-x})]^3$$

 

[Pietair]

Allright thanks, then I get:

$$0.5e^{3x}-0.5e^{-3x}=2e^{x}-2e^{-x}$$
Though I have no idea how to continue with this equation...

 

[phsopher]

How exactly did you arrive at that? It works for me.

 

[Pietair]

I made a mistake.

$$sinh^{3}(x) = 0.125[(e^{x}-e^{-x})]^3$$

This is not equal to:

$$sinh^{3}(x) = 0.125(e^{3x}-e^{-3x})$$

right?

 

[phsopher]

Remember:

$$(a - b)^3 = a^3 - 3 a^2 b + 3a b^2 - b^3$$

 

[Pietair]

Off course, thanks a lot!