Question

[Dark Killer]

1) Cho: \(\tan\alpha=\frac{1}{2}\). Tính \(\frac{\cos\alpha+\sin\alpha}{\cos\alpha-\sin\alpha}\)

2) Cho:  \(\cos\beta=2\sin\beta.\) Hãy tính: \(\sin\beta.\cos\beta\)

3)Chứng minh hệ thức:

a/ \(\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}\)

b/ \(\cot^2\alpha-\cos^2\alpha=\cot^2\alpha.\cos\alpha\)

Answer 1

[Cô Hoàng Huyền]

1. \(\frac{cos\alpha+sin\alpha}{cos\alpha-sin\alpha}=\frac{1+\frac{sin\alpha}{cos\alpha}}{1-\frac{sin\alpha}{cos\alpha}}=\frac{1+\frac{1}{2}}{1-\frac{1}{2}}=3\)

2. \(cos\beta=2sin\beta\Rightarrow cos^2\beta=4sin^2\beta\). Do \(cos^2\beta+sin^2\beta=1\Rightarrow5sin^2\beta=1\Rightarrow sin\beta=\frac{1}{\sqrt{5}}\)

\(\Rightarrow cos\beta=\frac{2}{\sqrt{5}}\). Vậy \(sin\beta.cos\beta=\frac{2}{5}\)

3. a. Nhân chéo ra được hệ thức \(sin^2\alpha+cos^2\alpha=1\)

b. Chú ý \(cot^2\alpha=\frac{cos^2\alpha}{sin^2\alpha}\)