Giải bài 1.14 trang 11

Đề bài

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 chi tiết

a) Ta có:

$\begin{array}{c}\sqrt{2}\cos \left(a+\frac{\pi }{4}\right)=\sqrt{2}\left(\cos a\cos \frac{\pi }{4}-\sin a\sin \frac{\pi }{4}\right)\\ =\sqrt{2}\left(\cos a.\frac{\sqrt{2}}{2}-\sin a.\frac{\sqrt{2}}{2}\right)\\ =\sqrt{2}.\cos a.\frac{\sqrt{2}}{2}-\sqrt{2}.\sin a.\frac{\sqrt{2}}{2}\\ =\cos a-\sin a.\end{array}$

b) Ta có:

$\begin{array}{c}VT=2\sin \left(a+\frac{\pi }{3}\right)\\ =2\left(\sin a\cos \frac{\pi }{3}+\cos a\sin \frac{\pi }{3}\right)\\ =2\left(\sin a.\frac{1}{2}+\cos a.\frac{\sqrt{3}}{2}\right)\\ =2\sin a.\frac{1}{2}+2\cos a.\frac{\sqrt{3}}{2}\\ =\sin a+\sqrt{3}\cos a.\end{array}$