L.H.S.

$=\log⁡(\sec ⁡x ) \\ =\log⁡ ( \dfrac 1\cos x ) \\ = \log⁡(\cos ⁡x )^{-1} \\ = -1 \log\cos ⁡x \\ =-1 \log⁡[1- \dfrac {x^2}{2!} - \dfrac{x^4}{4!} + \dfrac{x^6}{6!}…………)] \\ = (\dfrac{x^2}{2!} - \dfrac{x^4}{4!} + \dfrac{x^6}{6!}…………)+ \dfrac12 (\dfrac{x^2}{2!} - \dfrac{x^4}{4!} + \dfrac{x^6}{6!}…………)^2+ \dfrac13 (\dfrac{x^2}{2!} - \dfrac{x^4}{4!}…………)^3+⋯ \\ = \dfrac{x^2}{2!} - \dfrac{x^4}{4!} + \dfrac12 (\dfrac{x^2}{2!})^2+ \dfrac{x^6}{6!} -\dfrac12 (2 \dfrac{x^2}{2!} \dfrac{x^4}{4!} + \dfrac13 (\dfrac{x^2}{2!})^3+⋯ ) \\ =\dfrac{x^2}{2} + x^4 (\dfrac {-1}{24} + \dfrac18) + x^6 [\dfrac1{720} - \dfrac1{48} +\dfrac1{24}]+⋯ \\ =\dfrac{x^2}{2} + \dfrac{x^4}{12} + \dfrac{x^6}{45}+⋯ $

=R.H.S. Hence Proved.