Approximate value of tan $46°$ is ($1°=0.01745$ radians)

$$\begin{gathered} \text{ Let }\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)=\tan \mathrm{x} \\ \begin{array}{l}\mathrm{x}={45}^{\circ }=\frac{\mathrm{\pi }}{4}\\ \mathrm{\Delta x}=1^{\circ }=0.01745\mathrm{rad}\end{array} \\ \mathrm{y}=\mathrm{f}\left(\frac{\mathrm{\pi }}{4}\right)=\tan \frac{\mathrm{\pi }}{4}=1 \\ \text{ Let }\mathrm{dx}=\mathrm{\Delta x}=1^{\circ }=0.01745\mathrm{rad} \\ \mathrm{y}=\tan \mathrm{x} \\ \text{ Differentiating w.r.t. }\mathrm{x}\text{, we have } \\ \begin{array}{l}\frac{\mathrm{dy}}{\mathrm{dx}}={\sec }^2\mathrm{x}\\ \text{ at }\mathrm{x}=\frac{\mathrm{\pi }}{4}\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}={\sec }^2\frac{\mathrm{\pi }}{4}=2\\ \therefore \mathrm{dy}=\frac{\mathrm{dy}}{\mathrm{dx}}\times \mathrm{dx}=2\times 0.01745=0.0349\text{ rad. }[\because \mathrm{dx}=\mathrm{\Delta x}=0.01745]\\ \therefore \tan {46}^{\circ }=\mathrm{y}+\mathrm{\Delta y}=1+0.0349=1.0349\end{array} \end{gathered}$$