Bài 16. Chứng minh rằng: b) tan 20° - tan 40° +tan 80° = 33 c) tan 10 - tan 50 + tan 60 + tan 70 =2 3 d)tan 30 + tan 40 + tan 50 + tan 60 = $\frac{8 \sqrt{3} }{3}$

`b) tan 20° - tan 40° + tan 80° = 3sqrt(3)`

Ta có:

`VT = tan 80° - tan 40° + tan 20°`

`= tan(60° + 20°) - tan(60° - 20°) + tan 20°`

`= (tan 60° + tan 20°)/(1 - tan 60° tan 20°) - (tan 60° - tan 20°)/(1 + tan 60° tan 20°) + tan 20°`

`= (sqrt(3) + tan 20°)/(1 - sqrt(3) tan 20°) - (sqrt(3) - tan 20°)/(1 + sqrt(3) tan 20°) + tan 20°`

`= ((sqrt(3) + tan 20°)(1 + sqrt(3) tan 20°) - (sqrt(3) - tan 20°)(1 - sqrt(3) tan 20°)) / ((1 - sqrt(3) tan 20°)(1 + sqrt(3) tan 20°)) + tan 20°`

`= (sqrt(3) + 3tan 20° + tan 20° + sqrt(3)tan^2 20° - (sqrt(3) - 3tan 20° - tan 20° + sqrt(3)tan^2 20°)) / (1 - 3tan^2 20°) + tan 20°`

`= (8tan 20°) / (1 - 3tan^2 20°) + tan 20°`

`= (8tan 20° + tan 20°(1 - 3tan^2 20°)) / (1 - 3tan^2 20°)`

`= (9tan 20° - 3tan^3 20°) / (1 - 3tan^2 20°)`

`= 3 * (3tan 20° - tan^3 20°) / (1 - 3tan^2 20°)`

`= 3 * tan(3 * 20°) = 3 * tan(60°) = 3sqrt(3) = VP` `(đpcm)`

`c) tan 10° - tan 50° + tan 60° + tan 70° = 2sqrt(3)`

Ta có:

`VT = (tan 70° - tan 50°) + tan 10° + tan 60°`

`= tan(60° + 10°) - tan(60° - 10°) + tan 10° + tan 60°`

`= (tan 60° + tan 10°)/(1 - tan 60° tan 10°) - (tan 60° - tan 10°)/(1 + tan 60° tan 10°) + tan 10° + tan 60°`

`= (sqrt(3) + tan 10°)/(1 - sqrt(3) tan 10°) - (sqrt(3) - tan 10°)/(1 + sqrt(3) tan 10°) + tan 10° + sqrt(3)`

`= (8tan 10°) / (1 - 3tan^2 10°) + tan 10° + sqrt(3)`

`= (9tan 10° - 3tan^3 10°) / (1 - 3tan^2 10°) + sqrt(3)`

`= 3 * (3tan 10° - tan^3 10°) / (1 - 3tan^2 10°) + sqrt(3)`

`= 3 * tan(3 * 10°) + sqrt(3)`

`= 3 * tan(30°) + sqrt(3)`

`= 3 * 1/sqrt(3) + sqrt(3)`

`= sqrt(3) + sqrt(3) = 2sqrt(3) = VP` `(đpcm)`