tan(2x) = (2tanx) / (1 – tanx^2)
1
To prove the given equation:
We will start by using the double-angle formula for tangent, which states that
tan(2x) = (2tanx) / (1 – tan^2x)
So, we have
tan(2x) = (2tanx) / (1 – tan^2x)
Now, we can use the Pythagorean identity, which states that
1 + tan^2x = sec^2x
to simplify the denominator of the right-hand side of the equation:
tan(2x) = (2tanx) / (1 – tan^2x)
tan(2x) = (2tanx) / [(sec^2x) – 1]
tan(2x) = (2tanx) / (tan^2x + 1)
We can then substitute this expression for tan(2x) back into the original equation, to get
(2tanx) / (tan^2x + 1) = (2tanx) / (1 – tan^2x)
Multiplying both sides by (1 – tan^2x), we get
(2tanx) = (2tanx)(1 – tan^2x) / (tan^2x + 1)
Simplifying the right-hand side, we have
(2tanx) = (2tanx)(1 – tan^2x) / (tan^2x + 1)
(2tanx) = 2tanx – 2tan^3x
Rearranging terms, we get
2tan^3x = 0
Therefore, the only solution is
tanx = 0
which implies that
x = kπ
where k is an integer.
Hence, the solution to the initial equation tan(2x) = (2tanx) / (1 – tanx^2) is x = kπ, where k is an integer.
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