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$\left(1-\tan 348^{\circ}\right)\left(1+\cot 417^{\circ}\right)$ is equal to
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$\left(1-\tan 348^{\circ}\right)\left(1+\cot 417^{\circ}\right)$
$\begin{aligned} & \Rightarrow\left\{1-\tan \left(765^{\circ}-417^{\circ}\right\}\left\{1+\frac{1}{\tan 417^{\circ}}\right\}\right. \\ & \Rightarrow\left\{1-\left(\frac{\tan 765^{\circ}-\tan 417^{\circ}}{1+\tan 765^{\circ} \cdot \tan 417^{\circ}}\right)\right\} \cdot\left\{\frac{\tan 417^{\circ}+1}{\tan 417^{\circ}}\right\} \\ & \Rightarrow\left\{1-\left(\frac{1-\tan 417^{\circ}}{1+\tan 417^{\circ}}\right)\right\}\left\{\frac{1+\tan 417^{\circ}}{\tan 417^{\circ}}\right\} \\ & \Rightarrow\left\{\frac{1+\tan 417^{\circ}-1+\tan 417^{\circ}}{1+\tan 417^{\circ}}\right\}\left\{\frac{1+\tan 417^{\circ}}{\tan 417^{\circ}}\right\} \\ & =\frac{2 \tan 417^{\circ}}{1+\tan 417^{\circ}} \times \frac{1+\tan 417^{\circ}}{\tan 417^{\circ}}=2\end{aligned}$
$\begin{aligned} & \Rightarrow\left\{1-\tan \left(765^{\circ}-417^{\circ}\right\}\left\{1+\frac{1}{\tan 417^{\circ}}\right\}\right. \\ & \Rightarrow\left\{1-\left(\frac{\tan 765^{\circ}-\tan 417^{\circ}}{1+\tan 765^{\circ} \cdot \tan 417^{\circ}}\right)\right\} \cdot\left\{\frac{\tan 417^{\circ}+1}{\tan 417^{\circ}}\right\} \\ & \Rightarrow\left\{1-\left(\frac{1-\tan 417^{\circ}}{1+\tan 417^{\circ}}\right)\right\}\left\{\frac{1+\tan 417^{\circ}}{\tan 417^{\circ}}\right\} \\ & \Rightarrow\left\{\frac{1+\tan 417^{\circ}-1+\tan 417^{\circ}}{1+\tan 417^{\circ}}\right\}\left\{\frac{1+\tan 417^{\circ}}{\tan 417^{\circ}}\right\} \\ & =\frac{2 \tan 417^{\circ}}{1+\tan 417^{\circ}} \times \frac{1+\tan 417^{\circ}}{\tan 417^{\circ}}=2\end{aligned}$
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