tanα=3^X ,tanβ=3^(-X),α-β=π/6,则X=_____.
问题描述：
tanα=3^X ,tanβ=3^(-X),α-β=π/6,则X=_____.
答
tanα=3^X
tanβ=3^(-X)
α-β=π/6
tan(α-β) = (tanα-tanβ)/(1+tanα tanβ) = tan(π/6)
{3^x-3^(-x)} / {1+ 3^x *3^(-x)} = √3/3
{3^x-3^(-x)} / {1+ 1} = √3/3
3^x-3^(-x) = 2√3/3
3^x-1/3^x = 2√3/3
(3^x)^2 - 1 = 2√3/3 *3^x
(3^x)^2 - 2√3/3 *3^x = 1
(3^x - √3/3 )^2 = 1+1/3 = 4/3
3^x - √3/3 = ± 2√3 /3
3^x = √3/3 ± 2√3 /3
3^x = - 2√3 /3（舍去）
3^x = √3
x=1/2