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The general solution of $tan 2 x − tan x 1 + tan x tan 2 x = 1$ is:

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n π + π 4, ∀ n ∈ Z

n π ± π 4, ∀ n ∈ Z

∅

n π + π 6, ∀ n ∈ Z

answer is A.

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Detailed Solution

We have been given an equation

tan 2 x − tan x 1 + tan x tan 2 x = 1

.
We have to find the general solution of the given equation.
Now, let us consider the equation

tan 2 x − tan x 1 + tan x tan 2 x = 1

Now, we know that in trigonometry the tangent formula is given as

tan (A − B) = tan A − tan B 1 + tan A tan B

Now, when we compare the formula with the given equation we can write the equation as

⇒ tan 2 x − tan x 1 + tan 2 x tan x = 1 ⇒ tan (2 x − x) = 1 ⇒ tan x = 1

Now, we know that from trigonometric ratio table

tan π 4 = 1

Now, substituting the value in above equation we get

⇒ tan x = tan π 4

Or we can write as

x = π 4

Now, we know that if

θ

and

α

are not the multiples of

π 2

then we have

⇒ tan θ = tan α ⇒ θ = n π + α, ∀ n ∈ Z

Where,

Z

is the set of integers.
Now, we can write the obtained equation as

⇒ x = n π + π 4

So, the general solution of the above equation is

⇒ x = n π + π 4, ∀ n ∈ Z

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