Prove 2tana / 1-tan ^ 2A It is proved that 2tana / 1-tan ^ 2A = tan2a

Prove 2tana / 1-tan ^ 2A It is proved that 2tana / 1-tan ^ 2A = tan2a

Because there's a tangent and a formula
tan(A+B)=(tanA+tanB)/(1-tanA*tanB)
Let a = b = a
There are 2tana / (1-tan ^ 2a)

If 2tana = (1-tan ^ 2a), then tan2a=

tan2α=2tanα/(1-tan²α)=2tanα/2tanα=1

If Tan (a + b) = 2tana, it is proved that 3sinb = sin (2a + b)

Tan (a + b) = 2tanasin (a + b) * cosa = 2sina * cos (a + b) ① sin (a + b) * cosa Sina * cos (a + b) = Sina * cos (a + b) SINB = Sina * cos (a + b) sin (2a + B) = sin (a + b) * cosa + Sina * cos (a + b) (because ①) 3sinb = sin (2a + b)

It is proved that tan2acos4a-sin4a = (2tana) Tan ^ 2a-1

Left = sin2a (2cos2a ^ 2-1) - cos2a-2sin2acos2a = sin2a (2cos2a ^ 2-1-2cos2a ^ 2) - cos2a (common divisor is used in this step) = - sin2a / cos2a = - 2sinacosa \ (COSA ^ 2-sina ^ 2) (upper and lower divided by cosa ^ 2) = - 2tan2 \ (1-tana ^ 2) = 2tana \ (Tana ^ 2-1) = right

Find that Tan is the degree of an angle 2 times the root sign 3 of (15-4 times the root sign 3) Please come quickly!

23.22 degrees, which is calculated by a calculator

α∈ [0, π / 4], β∈ [0, π], Tan (α - β) = 1 / 2, Tan β = 1 / 7

tan(2α-β)=tan(2(α-β)+β)=(tan2(α-β) + tanβ)/(1-tan2(α-β)tanβ)
Because tan2 (α - β) = 2tan (α - β) / (1-tan ^ 2 (α - β)) = 4 / 3
So tan (2 α - β) = (4 / 3 + 1 / 7) / (1-4 / 3 * 1 / 7) = 31 / 17
α∈[0,π/4],β∈[0,π]
-β∈[-π,0],2α-β∈[-π,π/2],
2α-β=arc tan(31/17)

Is tan α / Tan β equal to tan2 α / tan2 β

Not necessarily. It's a very special situation
Counter example: α = 30 ° and β = 60 °
Tan α / Tan β is positive
Tan 2 α / Tan 2 β is negative

What is tan2 α equal to

tan2α=(2tanα)/(1-tan^2α)
Tan2 α = (2tan α) / (1 - (Tan α) ^ 2)

It is proved that Tan α - 1 / Tan α is equal to 2 / Tan 2 α

tanα-1/tanα
=sinα/cosα-cosα/sinα
=(sin²α-cos²α)/(sinαcosα)
=-cos2α/(1/2sin2α)
=-2cos2α/sin2α
=-2/tan2α

Who can tell me the whole process of tan2 α = 1 / 3 and how much Tan α equals

tan2α=(2tanα)/(1-(tanα)^2)=1/3
It can be solved
Tan α = - 3 + radical (10) or tan α = - 3-radical (10)