Question

Chứng minh rằng: 1/(tana+tanb) - 1/(Cota+Cotb) = Cot(a+b)

Answer 1

Lời giải:

Sử dụng các công thức sau:

\(\bullet \tan \alpha=\frac{1}{\cot \alpha}\)

\(\bullet \tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan\alpha.\tan \beta}\)

Ta có:

\(\text{VT}=\frac{1}{\tan a+\tan b}-\frac{1}{\cot a+\cot b}=\frac{1}{\tan a+\tan b}-\frac{1}{\frac{1}{\tan a}+\frac{1}{\tan b}}\)

\(=\frac{1}{\tan a+\tan b}-\frac{\tan a\tan b}{\tan a+\tan b}=\frac{1-\tan a\tan b}{\tan a+\tan b}\)

\(=\frac{1}{\frac{\tan a+\tan b}{1-\tan a\tan b}}=\frac{1}{\tan (a+b)}=\cot (a+b)=\text{VP}\)

Ta có đpcm.