Giải thích các bước giải:
\(\begin{array}{l}
c,\\
\tan a = \frac{1}{2} \Leftrightarrow \frac{{\sin a}}{{\cos a}} = \frac{1}{2} \Leftrightarrow \sin a = \frac{1}{2}\cos a\\
\tan b = \frac{1}{3} \Leftrightarrow \frac{{\sin b}}{{\cos b}} = \frac{1}{3} \Leftrightarrow \sin b = \frac{1}{3}\cos b\\
\tan \left( {a + b} \right) = \frac{{\sin \left( {a + b} \right)}}{{\cos \left( {a + b} \right)}} = \frac{{\sin a.\cos b + \cos a.\sin b}}{{\cos a.\cos b - \sin a.\sin b}}\\
= \dfrac{{\frac{1}{2}\cos a.\cos b + \cos a.\frac{1}{3}\cos b}}{{\cos a.\cos b - \frac{1}{2}\cos a.\frac{1}{3}\cos b}}\\
= \dfrac{{\frac{5}{6}\cos a.\cos b}}{{\frac{5}{6}\cos a.\cos b}}\\
= 1\\
\Rightarrow 0 < a,b < 90^\circ \Rightarrow 0 < a + b < 180^\circ \Rightarrow a + b = 45^\circ \\
b,\\
\tan \left( {\alpha + \frac{\pi }{4}} \right) = m\\
\Leftrightarrow \dfrac{{\sin \left( {\alpha + \frac{\pi }{4}} \right)}}{{\cos \left( {\alpha + \frac{\pi }{4}} \right)}} = m\\
\Leftrightarrow \dfrac{{\sin \alpha .\cos \frac{\pi }{4} + \cos \alpha .\sin \frac{\pi }{4}}}{{\cos \alpha .\cos \frac{\pi }{4} - \sin \alpha .\sin \frac{\pi }{4}}} = m\\
\Leftrightarrow \dfrac{{\sin \alpha + \cos \alpha }}{{\cos \alpha - \sin \alpha }} = m\\
\Leftrightarrow \dfrac{{\frac{{\sin \alpha }}{{\cos \alpha }} + 1}}{{1 - \frac{{\sin \alpha }}{{\cos \alpha }}}} = m\\
\Leftrightarrow \dfrac{{\tan \alpha + 1}}{{1 - \tan \alpha }} = m\\
\Leftrightarrow \tan \alpha + 1 = m - m.\tan \alpha \\
\Leftrightarrow \left( {m + 1} \right)\tan \alpha = m - 1\\
\Leftrightarrow \tan \alpha = \frac{{m - 1}}{{m + 1}}
\end{array}\)
