Question

Prove the identity \(coshx - sinhx = {e^{ - x}}\).

Short answer

The identity \(\cosh x - \sinh x = {e^{ - x}}\) is proved.

Step-by-step solution

Given identity

The identity is \(\cosh x - \sinh x = {e^{ - x}}\).

Formula of hyperbolic function and sine cosine functions

Hyperbolic function and sine cosine function:

\(\begin{array}{c}sinhx = \frac{{{e^x} - {e^{ - x}}}}{2}\\\cosh x = \frac{{{e^x} + {e^{ - x}}}}{2}\end{array}\)

Use the formula and substitute the value

Since the definition of the hyperbolic sine and cosine function are, \(\sin hx = \frac{{{e^x} - {e^{ - x}}}}{2}\) and \(\cosh x = \frac{{{e^x} + {e^{ - x}}}}{2}\).

\(\begin{array}{c}\cosh x - \sinh x = \frac{{{e^x} + {e^{ - x}}}}{2} - \frac{{{e^x} - {e^{ - x}}}}{2}\\ = \frac{{{e^x} + {e^{ - x}} - \left( {{e^x} - {e^{ - x}}} \right)}}{2}\\ = \frac{{2{e^{ - x}}}}{2}\\ = {e^{ - x}}\end{array}\)

Hence, the required identity \(\cosh x - \sinh x = {e^{ - x}}\) is proved.