Question

If $A-B=\frac{\mathrm{\pi }}{4},$ then (1 + tan A) (1 – tan B) = __________.

Answer 1

$$\begin{gathered} \mathrm{If}A-B=\frac{\mathrm{\pi }}{4} \\ \mathrm{then}\left(1+\tan A\right)\left(1-\tan B\right)=? \\ \mathrm{Since}A-B=\frac{\mathrm{\pi }}{4} \\ \mathrm{i}.\mathrm{e}.\tan \left(A-B\right)=\tan \frac{\mathrm{\pi }}{4} \\ \mathrm{i}.\mathrm{e}.\frac{\tan A-\tan B}{1+\tan A+\tan B}=1 \\ \mathrm{i}.\mathrm{e}.\tan A-\tan B=1+\tan A\tan B \\ \mathrm{i}.\mathrm{e}.\tan A-\tan B-\tan A\tan B=1 \\ i\mathit{.}e\mathit{.}\tan A\left(1-\tan B\right)-\tan B+1=1+1\left(\mathrm{adding}1\mathrm{both}\mathrm{sides}\right) \\ \mathrm{i}.\mathrm{e}.\left(\tan A+1\right)\left(1-\tan B\right)=2 \\ \mathrm{i}.\mathrm{e}.\mathrm{value}\mathrm{of}\left(1+\tan A\right)\left(1-\tan B\right)=2 \end{gathered}$$