Rút gọn các biểu thức sau sin(45 độ + a) - cos (45 độ + a) / sin(45 độ + a) + cos (45 độ + a)

Giải SBT Toán 11 Bài tập cuối chương 1 trang 25

Bài 1.55 trang 28 SBT Toán 11 Tập 1: Rút gọn các biểu thức sau

a) $\frac{\sin \left(45°+\alpha \right)-\cos \left(45°+\alpha \right)}{\sin \left(45°+\alpha \right)+\cos \left(45°+\alpha \right)}$;

b) $\frac{\sin 2\alpha +\sin \alpha }{1+\cos 2\alpha +\cos \alpha }$;

c) $\frac{1+\cos \alpha -\sin \alpha }{1-\cos \alpha -\sin \alpha }$;

d) $\frac{\sin \alpha +\sin 3\alpha +\sin 5\alpha }{\cos \alpha +\cos 3\alpha +\cos 5\alpha }$.

Lời giải:

a) $\frac{\sin \left(45°+\alpha \right)-\cos \left(45°+\alpha \right)}{\sin \left(45°+\alpha \right)+\cos \left(45°+\alpha \right)}$

$=\frac{\left(\sin 45°\cos \alpha +\cos 45°\sin \alpha \right)-\left(\cos 45°\cos \alpha -\sin 45°\sin \alpha \right)}{\left(\sin 45°\cos \alpha +\cos 45°\sin \alpha \right)+\left(\cos 45°\cos \alpha -\sin 45°\sin \alpha \right)}$

$=\frac{\left(\frac{\sqrt{2}}{2}\cos \alpha +\frac{\sqrt{2}}{2}\sin \alpha \right)-\left(\frac{\sqrt{2}}{2}\cos \alpha -\frac{\sqrt{2}}{2}\sin \alpha \right)}{\left(\frac{\sqrt{2}}{2}\cos \alpha +\frac{\sqrt{2}}{2}\sin \alpha \right)+\left(\frac{\sqrt{2}}{2}\cos \alpha -\frac{\sqrt{2}}{2}\sin \alpha \right)}$

$=\frac{\sqrt{2}\sin \alpha }{\sqrt{2}\cos \alpha }=\tan \alpha$.

b) $\frac{\sin 2\alpha +\sin \alpha }{1+\cos 2\alpha +\cos \alpha }$

$=\frac{2\sin \alpha \cos \alpha +\sin \alpha }{1+\left(2{\cos }^2\alpha -1\right)+\cos \alpha }$

$=\frac{2\sin \alpha \left(\cos \alpha +1\right)}{2\cos \alpha \left(\cos \alpha +1\right)}$

$=\frac{\sin \alpha }{\cos \alpha }=\tan \alpha$.

c) $\frac{1+\cos \alpha -\sin \alpha }{1-\cos \alpha -\sin \alpha }$

$=\frac{1+\left(2{\cos }^2\frac{\alpha }{2}-1\right)-2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}{1-\left(1-2{\sin }^2\frac{\alpha }{2}\right)-2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}$

$=\frac{2{\cos }^2\frac{\alpha }{2}-2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}{2{\sin }^2\frac{\alpha }{2}-2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}$

$=\frac{2\cos \frac{\alpha }{2}\left(\cos \frac{\alpha }{2}-\sin \frac{\alpha }{2}\right)}{2\sin \frac{\alpha }{2}\left(\sin \frac{\alpha }{2}-\cos \frac{\alpha }{2}\right)}$

$=-\frac{\cos \frac{\alpha }{2}}{\sin \frac{\alpha }{2}}=-\cot \frac{\alpha }{2}$.

d) $\frac{\sin \alpha +\sin 3\alpha +\sin 5\alpha }{\cos \alpha +\cos 3\alpha +\cos 5\alpha }$

$=\frac{\left(\sin 5\alpha +\sin \alpha \right)+\sin 3\alpha }{\left(\cos 5\alpha +\cos \alpha \right)+\cos 3\alpha }$

$=\frac{2\sin \frac{5\alpha +\alpha }{2}\cos \frac{5\alpha -\alpha }{2}+\sin 3\alpha }{2\cos \frac{5\alpha +\alpha }{2}\cos \frac{5\alpha -\alpha }{2}+\cos 3\alpha }$

$=\frac{2\sin 3\alpha \cos 2\alpha +\sin 3\alpha }{2\cos 3\alpha \cos 2\alpha +\cos 3\alpha }$

$=\frac{\sin 3\alpha \left(2\cos 2\alpha +1\right)}{\cos 3\alpha \left(2\cos 2\alpha +1\right)}$

$=\frac{\sin 3\alpha }{\cos 3\alpha }=\tan 3\alpha$.