Chứng minh rằng nếu cos(a + b) = 0 thì sin(a + 2b) = sin a.

Lời giải:

cos(a + b) = 0 Þ a + b = \(\frac{{\rm{\pi }}}{2} + {\rm{k\pi }}\,\,\left( {{\rm{k}} \in \mathbb{Z}} \right)\)

Û b = \( - {\rm{a}} + \frac{{\rm{\pi }}}{2} + {\rm{k\pi }}\,\,\left( {{\rm{k}} \in \mathbb{Z}} \right)\)

Û 2b = \( - 2{\rm{a}} + {\rm{\pi }} + {\rm{k2\pi }}\,\,\left( {{\rm{k}} \in \mathbb{Z}} \right)\)

sin (a + 2b)

= sin (a – 2a + p + k2p) \(\left( {{\rm{k}} \in \mathbb{Z}} \right)\)

= sin (–a + p + k2p) \(\left( {{\rm{k}} \in \mathbb{Z}} \right)\)

= sin (–a + p) \(\left( {{\rm{k}} \in \mathbb{Z}} \right)\)

= sin a (đpcm).