Power Reduction Formulas/Hyperbolic Cosine Cubed

Theorem

$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$

where $\cosh$ denotes hyperbolic cosine.

Proof 1

 $\displaystyle \cosh 3 x$ $=$ $\displaystyle 4 \cosh^3 x - 3 \cosh x$ Triple Angle Formula for Hyperbolic Cosine $\displaystyle \leadsto \ \$ $\displaystyle 4 \cosh^3 x$ $=$ $\displaystyle \cosh 3 x + 3 \cosh x$ rearranging $\displaystyle \leadsto \ \$ $\displaystyle \cosh^3 x$ $=$ $\displaystyle \dfrac {\cosh 3 x + 3 \cosh x} 4$ dividing both sides by $4$

$\blacksquare$

Proof 2

 $\displaystyle \cosh^3 x$ $=$ $\displaystyle \frac 1 {2^3} \paren {e^x + e^{-x} }^3$ Definition of Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \frac 1 8 \paren {e^{3x} + e^{-3x} + 3e^{x} + 3e^{-x} }$ $\displaystyle$ $=$ $\displaystyle \frac 1 4 \paren {\frac{ e^{3x} + e^{-3x} } 2} + \frac 3 4 \paren {\frac{e^{x} + e^{-x} } 2}$ $\displaystyle$ $=$ $\displaystyle \frac {\cosh 3x} 4 + \frac {3 \cosh x} 4$ Definition of Hyperbolic Cosine

$\blacksquare$

Proof 3

 $\displaystyle \cosh^3 x$ $=$ $\displaystyle \cos^3 i x$ Hyperbolic Cosine in terms of Cosine $\displaystyle$ $=$ $\displaystyle \frac {\map \cos {3 i x} + 3 \cos i x} 4$ Cube of Cosine $\displaystyle$ $=$ $\displaystyle \frac {\cosh 3 x + 3 \cosh x} 4$ Hyperbolic Cosine in terms of Cosine

$\blacksquare$