Question

To prove the identity \(cosh( - x) = coshx\).

Short answer

The identity \(\cosh ( - x) = \cosh x\) is proved.

Step-by-step solution

Given identity

The identity is \(\cosh ( - x) = \cosh x\).

Formula of hyperbolic function

Hyperbolic function:\(coshx = \frac{{{e^x} + {e^{ - x}}}}{2}\)

Use the formula and substitute the value

Use the formula of the hyperbolic function, \(\cosh x = \frac{{{e^x} + {e^{ - x}}}}{2}\).

Substitute, \(x = - x\).

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

Note that, the function \(f(x)\) is even function if \(f( - x) = f(x)\).

Thus, \(\cos hx\) is even function.

Hence, the required identity \(\cosh ( - x) = \cosh x\) is proved.