Giá trị của tan a = √ 7/3 .

a) Có \({\sin ^2}a = 1 - {\cos ^2}a = 1 - {\left( {\frac{3}{4}} \right)^2} = \frac{7}{{16}}\) mà sina > 0 nên \(\sin a = \frac{{\sqrt 7 }}{4}\).

Do đó \(\tan a = \frac{{\sin a}}{{\cos a}} = \frac{{\sqrt 7 }}{4}:\frac{3}{4} = \frac{{\sqrt 7 }}{3}\).

b) Có \({\cos ^2}b = 1 - {\sin ^2}b = 1 - {\left( {\frac{3}{5}} \right)^2} = \frac{{16}}{{25}}\) mà cosb < 0 Þ \(\cos b =  - \frac{4}{5}\).

Do đó \(\cot b = \frac{{\cos b}}{{\sin b}} =  - \frac{4}{5}:\frac{3}{5} =  - \frac{4}{3}\).

c) cos2a + cos2b = 2cos2a – 1 + 2cos2b – 1 = \(2.{\left( {\frac{3}{4}} \right)^2} + 2.{\left( {\frac{{ - 4}}{5}} \right)^2} - 2 = \frac{{81}}{{200}} \notin \left( {\frac{1}{2};1} \right)\).

d) cos(a + b) = cosacosb – sinasinb = \(\frac{3}{4}.\left( {\frac{{ - 4}}{5}} \right) - \frac{{\sqrt 7 }}{4}.\frac{3}{5} = \frac{{ - 12 - 3\sqrt 7 }}{{20}} \notin \left( { - \frac{1}{2}; - \frac{1}{3}} \right)\).

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