Question

Prove the identity.

23. \({\left( {{\bf{cosh}}\,x + {\bf{sinh}}\,x} \right)^n} = {\bf{cosh}}\,nx + {\bf{sinh}}\,nx\)

(n any real number)

Short answer

The given identity is true.

Step-by-step solution

Step 1:Solve the expression \({\left( {{\bf{cosh}}\,x + {\bf{sinh}}\,x} \right)^n}\)

Solve the expression \(\tanh \left( {\ln x} \right)\) using hyperbolic relations.

\(\begin{aligned}{\left( {\cosh x + \sinh x} \right)^n} &= {\left( {\frac{1}{2}\left( {{e^x} + {e^{ - x}}} \right) + \frac{1}{2}\left( {{e^x} - {e^{ - x}}} \right)} \right)^n}\\ &= {\left( {\frac{{2{e^x}}}{2}} \right)^n}\\ &= {\left( {{e^x}} \right)^n}\end{aligned}\)

Solve the equation in step 1

Simplify the equation \({\left( {\cosh x + \sinh x} \right)^n} = {\left( {{e^x}} \right)^n}\) using the power rule.

\(\begin{aligned}{\left( {\cosh x + \sinh x} \right)^n} &= {e^{nx}}\\ &= \cosh nx + \sinh nx\end{aligned}\)

Hence proved, the given identity is true.