Question 19

$\begin{aligned} \tan{A}+1&=\dfrac{\sin{A}}{\cos{A}}+1 \\ \\ &=\dfrac{\sin{A}+\cos{A}}{\cos{A}} \\ \\ &=\sqrt{2}\sin\left(A+\frac{\pi}{4}\right)\sec{A} &&\{\sin{A}+\cos{A}=\sqrt{2}\left(\sin{A}\cos{\frac{\pi}{4}}+\cos{A}\sin{\frac{\pi}{4}}\right)\} \\ \\ \therefore\enspace\tan{A}+1&\leq\sqrt{2}\sec{A}\quad....\enspace(1) &&\{\text{as }\sin{(A+\frac{\pi}{4})} \leq 1 \quad\text{for all}\quad 0 < A < \frac{\pi}{2} \} \\ \\ \text{and similarly}\quad\tan{B}+1&\leq\sqrt{2}\sec{B}\quad....\enspace(2) \\ \\ \llap{\text{Adding inequalities (1) an}} \rlap{\text{d (2) gives}} \\ \\ \tan{A}+\tan{B}+2&\leq\sqrt{2}(\sec{A} + \sec{B}) \end{aligned}$