How to derive the identity for hyperbolic sine?

How to derive the identity for hyperbolic sine?

The identity for hyperbolic sine is derived using the exponential function and its properties.

The hyperbolic sine function is defined as:

sinh(x) = (e^x - e^(-x))/2

To derive the identity, we start with the definition of sinh(x) and multiply the numerator and denominator by e^x:

sinh(x) = (e^x - e^(-x))/2 * e^x/e^x

= (e^(2x) - 1)/(2e^x)

Next, we use the identity e^(2x) = (e^x)^2 to rewrite the numerator:

sinh(x) = ((e^x)^2 - 1)/(2e^x)

= (e^x + 1)(e^x - 1)/(2e^x)

= (e^x/2 - e^(-x)/2)(e^x/2 + e^(-x)/2)

= (e^x/2)^2 - (e^(-x)/2)^2

= (e^x/2)^2 - (1 - e^x/2)^2

= 1 - e^x/2 * (1 - e^x/2)

= 1 - (1 - e^x)/2

= e^x/2

Therefore, we have derived the identity:

sinh(x) = (e^x - e^(-x))/2 = e^x/2 - e^(-x)/2 = (e^x - 1)/(2e^(-x))