Build a shifted-tangent expansion and measure its error

$y^\prime=\sec^2(\pi/4+x)=1+y^2$, so $y^{\prime\prime}=2yy^\prime$. Further differentiation gives $y^{\prime\prime\prime}=2yy^{\prime\prime}+2(y^\prime)^2$ and $y^{(4)}=2yy^{\prime\prime\prime}+6y^\prime y^{\prime\prime}$.

At zero, $y=1$, $y^\prime=2$, $y^{\prime\prime}=4$, $y^{\prime\prime\prime}=16$ and $y^{(4)}=80$. Therefore $y=1+2x+4x^2/2!+16x^3/3!+80x^4/4!+O(x^5)=1+2x+2x^2+\tfrac83x^3+\tfrac{10}3x^4+O(x^5)$.

At $x=0.1$, the fourth-degree polynomial equals $1.223$. Relative to the supplied value, the percentage error is $\dfrac{1.22304888-1.223}{1.22304888}\times100\%\approx0.00399657\%\approx0.004\%$.