Question

[Le van a]

Chứng minh rằng:

\(\sqrt{\dfrac{1+cos\alpha}{1-cos\alpha}}-\sqrt{\dfrac{1-cos\alpha}{1+cos\alpha}}=2cot\alpha\left(0< \alpha< \dfrac{\pi}{2}\right)\).

Answer 1

[Akai Haruma]

Lời giải:

Do \(0< a< \frac{\pi}{2}\Rightarrow \sin a>0\)

Ta có:

\(\sqrt{\frac{1+\cos a}{1-\cos a}}-\sqrt{\frac{1-\cos a}{1+\cos a}}=\frac{(1+\cos a)-(1-\cos a)}{\sqrt{(1-\cos a)(1+\cos a)}}\)

\(=\frac{2\cos a}{\sqrt{1-\cos ^2a}}=\frac{2\cos a}{\sqrt{\sin ^2a}}=\frac{2\cos a}{\sin a}\)

\(=2.\frac{\cos a}{\sin a}=2\cot a\)

Ta có đpcm.