Proving the Equation for Tan(2x): Steps and Solution | Math Tutorial

tan(2x) = (2tanx) / (1 – tanx^2)

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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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