Question

tan 10 - tan 50 +tan 70 with trigonometry
answer is sqrt(3)
but i cant solve it

Answer 1

\[\tan 10 - \tan 50 +\tan 70 = \sqrt{3}\]


IS THIS WHAT YOUR SAYING?

Answer 2

ya

Answer 3

tan70 = tan(60+10)
use the identity :
tan(a+b) = (tan(a) + tan(b))/(1-tan(a)tan(b))
so, tan70 = (tan60+tan10)/(1-tan60tan10)
tan70 = (sqrt(3)+tan(10))/(1-sqrt(3)tan10) ... (1)

tan50 = tan(60-10)
use the identity :
tan(a-b) = (tan(a) - tan(b))/(1+tan(a)tan(b))
so, tan50 = (tan60-tan10)/(1+tan60tan10)
tan50 = (sqrt(3)-tan(10))/(1+sqrt(3)tan10) ... (2)

therefore, the value of tan70 - tan50 =
(sqrt(3)+tan(10))/(1-sqrt(3)tan10) - (sqrt(3)+tan(10))/(1-sqrt(3)tan10)
if that simplied becomes,
we get tan70 - tan50 = 8tan10/(1-3tan^2 10)

Answer 4

now, look at the original question above is , sin10-sin50+tan70 which similar with tan70 - tan50 + tan10. (the value of tan70-tan50 already done)
so, tan70 - tan50 + tan10
= 8tan10/(1-3tan^2 10) + tan10
= (8tan10+tan10-3tan^3 10)/(1-3tan^2 10)
= (9tan10-3tan^3 10)/(1-3tan^2 10)
= 3 (3tan10 - tan^3 10)/(1-3tan^2 10)

Answer 5

then, remember that there is a formula in trigono :
tan(3x) = (3tan(x) - tan^3 (x))/(1 - 3tan^2 (x))

Answer 6

therefore, the calculation above can be simplied becomes
3 (3tan10 - tan^3 10)/(1-3tan^2 10)
= 3 tan(3*10)
= 3 tan30
= 3 * 1/3 sqrt(3)
= sqrt(3)

QED ^_^