MHB Series Expansion: Show Sin^Cos x = x + O(x^3)

[polygamma]

Show that for small positive $x$, $$\left( \sin x \right)^{\cos x} = x -\left( 3 \log x + 1\right) \frac{x^{3}}{3!} + \Big( 15 \log^{2} x + 15 \log x + 11 \Big) \frac{x^{5}}{5!} + \mathcal{O}(x^{7})$$

 

[polygamma]

Spoiler

$$ \large (\sin x)^{\cos x} = e^{\cos (x) \log (\sin x) } $$

$$ \large = e^{\cos (x) [\log (x- \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \ldots)]}$$

$$ \large = e^{\cos (x) [\log x + \log (1-\frac{x^{2}}{3!} + \frac{x^{4}}{5!} + \ldots)]}$$

$$ \large = e^{\cos (x) [ \log x -(\frac{x^{2}}{3!} - \frac{x^{4}}{5!} + \frac{x^{4}}{2(3!)^{2}} + \ldots)]}$$

$$ \large = e^{\cos (x) (\log x - \frac{x^{2}}{3!} - \frac{x^{4}}{180} + \ldots)} $$

$$ \large =e^{(1- \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \ldots )(\log x - \frac{x^{2}}{3!} - \frac{x^{4}}{180} + \ldots )}$$

$$ \large = e^{\log x - \frac{x^{2}}{3!} - \frac{x^{4}}{180} - \frac{x^{2}}{2!} \log x + \frac{x^{4}}{2!(3!)} + \frac{x^{4}}{4!} \log x + \ldots}$$

$$ = \large x e^{-\frac{x^{2}}{3!} - \frac{x^{2}}{2!} \log x} e^{\frac{7x^{4}}{90}+ \frac{x^{4}}{4!} \log x} \times \cdots$$

$$ =x \left( 1 - \frac{x^{2}}{3!} - \frac{x^{2}}{2!} \log x + \frac{1}{2!} \left(\frac{x^{2}}{3!} + \frac{x^{2}}{2!} \log x \right)^{2} + \ldots \right) \left( 1 + \frac{7 x^{4}}{90} + \frac{x^{4}}{4!} \log x + \ldots \right) \times \cdots$$

$$ = x \left(1 + \frac{7x^{4}}{90} + \frac{x^{4}}{4!} \log x - \frac{x^{2}}{3!} - \frac{x^{2}}{2!} \log x + \frac{x^{4}}{2!(3!)^{2}} + \frac{2 x^{4}}{(2!)^{2}(3!)} \log x+ \frac{x^{4}}{2!(2!)^{2}} \log^{2} x + \ldots \right) $$

$$ = x - x \left( \frac{x^{2}}{3!} + \frac{x^{2}}{2!} \log x \right) + x \left(\frac{11 x^{4}}{120} + \frac{x^{4}}{8} \log x + \frac{x^{4}}{8}\log^{2} x \right) + \ldots $$

$$ = x - \left( 3 \log x +1 \right) \frac{x^{3}}{3!} + \left(15 \log^{2} x + 15 \log x + 11 \right) \frac{x^{5}}{5!} + \ldots $$