How can we prove that tan(2x) = 2tan(x) / (1 – tan^2(x))?

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To prove the identity tan(2x) = \frac{2tan(x)}{1 – tan^2(x)}, we can start by using the double angle formula for tangent. The double angle formula states:

tan(2x) = \frac{2tan(x)}{1 – tan^2(x)}

Here’s a step-by-step breakdown of the proof:

  1. Recall the definitions: The tangent function can be expressed as the ratio of sine and cosine functions:

    2. tan(x) = \frac{sin(x)}{cos(x)}

  2. Use the sine and cosine double angle formulas: We know:

    3. sin(2x) = 2sin(x)cos(x)

    cos(2x) = cos^2(x) – sin^2(x)

  3. Express tan(2x) using sin(2x) and cos(2x):

    4. tan(2x) = \frac{sin(2x)}{cos(2x)} = \frac{2sin(x)cos(x)}{cos^2(x) – sin^2(x)}

  4. Rewrite sin and cos in terms of tan:

    5. Recall that sin(x) = \frac{tan(x)cos(x)}{1 + tan^2(x)} and cos(x) = \frac{1}{\sqrt{1 + tan^2(x)}}.

  5. Substituting tan(x):

    6. From above, expressing sin and cos in terms of tan leads to:

    sin(x) = \frac{tan(x)}{\sqrt{1 + tan^2(x)}}

    substituting back we derive:
    \frac{2tan(x)\frac{1}{\sqrt{1 + tan^2(x)}}(\sqrt{1 + tan^2(x)})}{\frac{1 – tan^2(x)}{1 + tan^2(x)}}.

  6. Simplification: When everything is simplified, we arrive back to:

    7. tan(2x) = \frac{2tan(x)}{1 – tan^2(x)}.

Thus, by verifying the derivations step by step, we confirm that the identity holds:

tan(2x) = \frac{2tan(x)}{1 – tan^2(x)}

This completes our proof!

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