Given,
f(x)={\begin{matrix}\frac{{\mathrm{log}}_{e}(1-x+{x}^{2})+{\mathrm{log}}_{e}(1+x+{x}^{2})}{secx-\mathrm{cos}x}, & x\in (\frac{-\pi }{2},\frac{\pi }{2})-{0} \\ k & ,x=0\end{matrix}
⇒x→0lim1−cos2x(ln(1+x2+x4))cosx=k
Taking L.H.S=x→0lim1−cos2x(ln(1+x2+x4))cosx
=x→0lim(x2sin2x)x2(x2+x4ln(1+x2+x4))x2(1+x2)cosx
=x→0lim(x2sin2x)(x2+x4ln(1+x2+x4))(1+x2)cosx
=(1)(1)(1+0)1=1
Now equating with R.H.S we get, k=1