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Câu hỏi: Rút gọn các biểu thức
a) $\frac{\sin 2 \alpha+\sin \alpha}{1+\cos 2 \alpha+\cos \alpha}$;
b) $\frac{4 \sin ^{2} \alpha}{1-\cos ^{2} \frac{\alpha}{2}}$
c) $\frac{1+\cos \alpha-\sin \alpha}{1-\cos \alpha-\sin \alpha}$
d) $\frac{1+\sin \alpha-2 \sin ^{2}\left(45^{\circ}-\frac{\alpha}{2}\right)}{4 \cos \frac{\alpha}{2}}$
a) $\frac{\sin 2 \alpha+\sin \alpha}{1+\cos 2 \alpha+\cos \alpha}$;
b) $\frac{4 \sin ^{2} \alpha}{1-\cos ^{2} \frac{\alpha}{2}}$
c) $\frac{1+\cos \alpha-\sin \alpha}{1-\cos \alpha-\sin \alpha}$
d) $\frac{1+\sin \alpha-2 \sin ^{2}\left(45^{\circ}-\frac{\alpha}{2}\right)}{4 \cos \frac{\alpha}{2}}$
a)
$\begin{aligned} \frac{\sin 2 \alpha+\sin \alpha}{1+\cos 2 \alpha+\cos \alpha} &=\frac{\sin \alpha(2 \cos \alpha+1)}{2 \cos ^{2} \alpha+\cos \alpha} \\ &=\frac{\sin \alpha(2 \cos \alpha+1)}{\cos \alpha(2 \cos \alpha+1)}=\tan \alpha \end{aligned}$
b) $\frac{4 \sin ^{2} \alpha}{1-\cos ^{2} \frac{\alpha}{2}}=\frac{16 \sin ^{2} \frac{\alpha}{2} \cos ^{2} \frac{\alpha}{2}}{\sin ^{2} \frac{\alpha}{2}}=16 \cos ^{2} \frac{\alpha}{2}$
c)
$\begin{aligned} \frac{1+\cos \alpha-\sin \alpha}{1-\cos \alpha-\sin \alpha} &=\frac{2 \cos ^{2} \frac{\alpha}{2}-2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}}{2 \sin ^{2} \frac{\alpha}{2}-2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}} \\ &=\frac{2 \cos \frac{\alpha}{2}\left(\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2}\right)}{2 \sin \frac{\alpha}{2}\left(\sin \frac{\alpha}{2}-\cos \frac{\alpha}{2}\right)}=-\cot \frac{\alpha}{2} . \end{aligned}$
d)
$1+\sin \alpha-2 \sin ^{2}\left(45^{\circ}-\frac{\alpha}{2}\right)=\frac{\sin \alpha+\cos \left(90^{\circ}-\alpha\right)}{4 \cos \frac{\alpha}{2}}$
$=\frac{\sin \alpha+\sin \alpha}{4 \cos \frac{\alpha}{2}}=\frac{4 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}}{4 \cos \frac{\alpha}{2}}=\sin \frac{\alpha}{2}$
$\begin{aligned} \frac{\sin 2 \alpha+\sin \alpha}{1+\cos 2 \alpha+\cos \alpha} &=\frac{\sin \alpha(2 \cos \alpha+1)}{2 \cos ^{2} \alpha+\cos \alpha} \\ &=\frac{\sin \alpha(2 \cos \alpha+1)}{\cos \alpha(2 \cos \alpha+1)}=\tan \alpha \end{aligned}$
b) $\frac{4 \sin ^{2} \alpha}{1-\cos ^{2} \frac{\alpha}{2}}=\frac{16 \sin ^{2} \frac{\alpha}{2} \cos ^{2} \frac{\alpha}{2}}{\sin ^{2} \frac{\alpha}{2}}=16 \cos ^{2} \frac{\alpha}{2}$
c)
$\begin{aligned} \frac{1+\cos \alpha-\sin \alpha}{1-\cos \alpha-\sin \alpha} &=\frac{2 \cos ^{2} \frac{\alpha}{2}-2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}}{2 \sin ^{2} \frac{\alpha}{2}-2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}} \\ &=\frac{2 \cos \frac{\alpha}{2}\left(\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2}\right)}{2 \sin \frac{\alpha}{2}\left(\sin \frac{\alpha}{2}-\cos \frac{\alpha}{2}\right)}=-\cot \frac{\alpha}{2} . \end{aligned}$
d)
$1+\sin \alpha-2 \sin ^{2}\left(45^{\circ}-\frac{\alpha}{2}\right)=\frac{\sin \alpha+\cos \left(90^{\circ}-\alpha\right)}{4 \cos \frac{\alpha}{2}}$
$=\frac{\sin \alpha+\sin \alpha}{4 \cos \frac{\alpha}{2}}=\frac{4 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}}{4 \cos \frac{\alpha}{2}}=\sin \frac{\alpha}{2}$