What is the formula for the tangent of a double angle, tan(2x)?

The formula for the tangent of a double angle, tan(2x), is derived from the sine and cosine double angle identities. It is expressed as:

[

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

]

Using the double angle identities for sine and cosine:

[

\sin(2x) = 2\sin(x)\cos(x)

]

[

\cos(2x) = \cos^2(x) - \sin^2(x)

]

We can rewrite the tangent in terms of sine and cosine:

[

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

]

Next, recall that (\tan(x) = \frac{\sin(x)}{\cos(x)}). Thus, we have:

[

\sin(x) = \tan(x)\cos(x) \quad \text{and} \quad \cos(x) = \frac{1}{\sqrt{1 + \tan^2(x)}}

]

Substituting these back into our equation for tangent gives:

[

\tan(2x