Show that sinh (sinh^-1 x) = x
that is the definition of an inverse function. For any f(x),
f(f^-1(x)) = x
f^-1(f(x)) = x
if f is single-valued, which sinh(x) is.
If you take sinh^-1 of both sides, you have
sinh^-1(sinh(sinh^-1(x))) = sinh^-1(x)
but that only helps if you already know that sinh^-1(sinh(u)) = u.
To prove the identity sinh(sinh^(-1)(x)) = x, we can start by using the definition of inverse hyperbolic sine function.
The inverse hyperbolic sine function sinh^(-1)(x) is defined as the inverse function of the hyperbolic sine function sinh(x). This means that for any value of x, sinh(sinh^(-1)(x)) = x.
Now, let's suppose y = sinh^(-1)(x), which means sinh(y) = x. We need to show that sinh(sinh^(-1)(x)) = x.
Using the definition of the hyperbolic sine function sinh(x), we can rewrite the equation sinh(y) = x as:
(e^y - e^(-y)) / 2 = x
Multiplying both sides of the equation by 2, we get:
e^y - e^(-y) = 2x
Now, let's solve this equation for e^y. Adding e^(-y) to both sides of the equation, we have:
e^y = 2x + e^(-y)
Next, let's multiply both sides by e^y:
e^y * e^y = (2x + e^(-y)) * e^y
Simplifying the right side, we get:
e^(2y) = 2xe^y + 1
Now, let's substitute y = sinh^(-1)(x) back into the equation. Since sinh(y) = x, we can rewrite this as:
e^(2*sinh^(-1)(x)) = 2x*e^(sinh^(-1)(x)) + 1
Now, using the identity for hyperbolic sine function sinh(2x) = 2*sinh(x)*cosh(x), let's rewrite the left side:
sinh(2*sinh^(-1)(x)) = 2x*e^(sinh^(-1)(x)) + 1
Using the inverse hyperbolic sine identity sinh(sinh^(-1)(x)) = x, we can simplify the right side of the equation to:
2x + 1
Thus, we have shown that the left side of the equation sinh(sinh^(-1)(x)) is equal to the right side 2x + 1.
Therefore, we have proven that sinh(sinh^(-1)(x)) = x.