Nếu \(\alpha  + \beta  + \gamma  = \frac{\pi }{2}\) và \(\cot \alpha  + \cot \gamma  = 2\cot \beta \) thì \(\cot \alpha .\cot \gamma \) bằng

Vì \(a,b\) là các góc nhọn nên \(\cos a > 0,\cos b > 0\).

Ta có \(\cos a = \sqrt {1 - {{\sin }^2}a}  = \frac{{15}}{{17}} \Rightarrow \tan a = \frac{{\sin a}}{{\cos a}} = \frac{8}{{15}}\);

          \(\cos b = \sqrt {\frac{1}{{1 + {{\tan }^2}b}}}  = \frac{{12}}{{13}} \Rightarrow \sin b = \cos b\tan b = \frac{5}{{13}}{\rm{. }}\)

Khi đó, \(\sin \left( {a - b} \right) = \sin a\cos b - \cos a\sin b = \frac{8}{{17}} \cdot \frac{{12}}{{13}} - \frac{{15}}{{17}} \cdot \frac{5}{{13}} = \frac{{21}}{{221}}\).

          \(\cos \left( {a + b} \right) = \cos a\cos b - \sin a\sin b = \frac{{15}}{{17}} \cdot \frac{{12}}{{13}} - \frac{8}{{17}} \cdot \frac{5}{{13}} = \frac{{140}}{{221}}\)

          \(\tan \left( {a + b} \right) = \frac{{\tan a + \tan b}}{{1 - \tan a\tan b}} = \frac{{\frac{8}{{15}} + \frac{5}{{12}}}}{{1 - \frac{8}{{15}} \cdot \frac{5}{{12}}}} = \frac{{171}}{{140}}.\)

Đáp án:       a) Đúng,      b) Đúng,      c) Sai,                    d) Sai.

Ta có \(\tan \alpha  = \tan \left( {\widehat {BAD} - \widehat {CAD}} \right) = \frac{{\tan \widehat {BAD} - \tan \widehat {CAD}}}{{1 + \tan \widehat {BAD}\tan \widehat {CAD}}} = \frac{{\frac{{15}}{{12}} - \frac{9}{{12}}}}{{1 + \frac{{15}}{{12}} \cdot \frac{9}{{12}}}} = \frac{8}{{31}}.\)

Vì vậy \(\alpha  \approx 14,5^\circ \).

Đáp án: \(14,5\).