To prove that tan(2x) = (2tan(x)) / (1 – tan^2(x)), we start with the double angle formula for tangent. The tangent of a double angle, such as tan(2x), can be expressed in terms of the tangent of the angle x.
The double angle formula for tangent states:
tan(2x) = rac{2tan(x)}{1 – tan^2(x)}
To derive this, we can use the sine and cosine definitions of tangent:
- We know that tan(x) = rac{sin(x)}{cos(x)}
- Therefore, we can express sin(2x) and cos(2x):
- sin(2x) = 2sin(x)cos(x)
- cos(2x) = cos^2(x) – sin^2(x)
Next, we can substitute these into the definition of tangent:
tan(2x) = rac{sin(2x)}{cos(2x)} = rac{2sin(x)cos(x)}{cos^2(x) – sin^2(x)}
Now we can express sin^2(x) in terms of tan(x):
- sin^2(x) = rac{tan^2(x)}{1 + tan^2(x)}
- cos^2(x) = rac{1}{1 + tan^2(x)}
Substituting these back into our equation gives:
tan(2x) = rac{2 rac{tan(x)}{1 + tan^2(x)} rac{1}{1 + tan^2(x)}}{ rac{1}{1 + tan^2(x)} – rac{tan^2(x)}{1 + tan^2(x)} }
Now simplify the denominator:
tan(2x) = rac{2tan(x)}{1 – tan^2(x)}
This shows that the equation holds true, hence we have proven that:
tan(2x) = rac{2tan(x)}{1 – tan^2(x)}