\(\sin (a - b) = \sin a\cos b - \cos a\sin b\); \(\sin (a + b) = \sin a\cos b + \cos a\sin b\); \(\cos (a - b) = \cos a\cos b + \sin a\sin b\); \(\cos (a + b) = \cos a\cos b - \sin a\sin b\); \(\tan (a - b) = \frac{{\tan a - \tan b}}{{1 + \tan a\tan b}}\); \(\tan (a + b) = \frac{{\tan a + \tan b}}{{1 - \tan a\tan b}}\).

1) Không dùng máy tính, hãy tính:

a) \(\cos {75^o}\); b) \(\tan \frac{\pi }{2}\).

Giải:

a) \(\cos {75^o} = \cos ({45^o} + {30^o}) = \cos {45^o}\cos {30^o} - \sin {45^o}\sin {30^o}\) \( = \frac{{\sqrt 2 }}{2}.\frac{{\sqrt 3 }}{3} - \frac{{\sqrt 2 }}{2}.\frac{1}{2} = \frac{{\sqrt 6  - \sqrt 2 }}{4}\). b) \(\tan \frac{\pi }{{12}} = \tan \left( {\frac{\pi }{3} - \frac{\pi }{4}} \right) = \frac{{\tan \frac{\pi }{3} - \tan \frac{\pi }{4}}}{{1 + \tan \frac{\pi }{3}.\tan \frac{\pi }{4}}} = \frac{{\sqrt 3  - 1}}{{1 + \sqrt 3 }} = 2 - \sqrt 3 \).

2) Chứng minh rằng \(\sin x + \cos x = \sqrt 2 \sin \left( {x + \frac{\pi }{4}} \right)\).

Giải:

Ta có \(\sqrt 2 \sin \left( {x + \frac{\pi }{4}} \right) = \sqrt 2 \left( {\sin x\cos \frac{\pi }{4} + \cos x\sin \frac{\pi }{4}} \right)\) \( = \sqrt 2 \left( {\sin x\frac{{\sqrt 2 }}{2} + \cos x\frac{{\sqrt 2 }}{2}} \right) = \sin x + \cos x\) (đpcm).