Question

Prove the identity \(sinh( - x) = - sinhx\).

Short answer

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

Step-by-step solution

Given identity

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

Formula of hyperbolic function

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

Use the formula and substitute the value

The formula of the hyperbolic function, \(\sin hx = \frac{{{e^x} - {e^{ - x}}}}{2}\).

Substitute, \(x = - x\).

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

Note that, the function \(f(x)\) is called an odd function if \(f( - x) = - f(x)\).

That is, \(\sinh x\) is an odd function.

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