Chứng minh rằng:

a) \(\cos a - \sin a = \sqrt 2 \cos \left( {a + \frac{\pi }{4}} \right)\);

b) \(\sin a + \sqrt 3 \cos a = 2\sin \left( {a + \frac{\pi }{3}} \right)\).

Lời giải

a) \(VP = \sqrt 2 \cos \left( {a + \frac{\pi }{4}} \right) = \sqrt 2 \left( {\cos a\cos \frac{\pi }{4} - \sin a\sin \frac{\pi }{4}} \right)\)\( = \sqrt 2 \left( {\frac{{\sqrt 2 }}{2}\cos a - \frac{{\sqrt 2 }}{2}\sin a} \right)\)

\( = \sqrt 2 .\frac{{\sqrt 2 }}{2}\left( {\cos a - \sin a} \right) = \cos a - \sin a = VT\).

b) \(VP = 2\sin \left( {a + \frac{\pi }{3}} \right) = 2\left( {\sin a\cos \frac{\pi }{3} + \cos a\sin \frac{\pi }{3}} \right)\)

\( = 2\left( {\frac{1}{2}\sin a + \frac{{\sqrt 3 }}{2}\cos a} \right)\)\( = \sin a + \sqrt 3 \cos a = VT\).