Simplifying the Expression: 1 - tan^2(x) / 1 + tan^2(x)
In this article, we will explore the simplification of the algebraic expression:
$\frac{1 - \tan^2(x)}{1 + \tan^2(x)}$
Using the Pythagorean Identity
To simplify this expression, we can use the Pythagorean identity:
$\sin^2(x) + \cos^2(x) = 1$
We can rewrite this identity as:
$\tan^2(x) + 1 = \frac{1}{\cos^2(x)}$
Substituting the Identity
Now, let's substitute this identity into the original expression:
$\frac{1 - \tan^2(x)}{1 + \tan^2(x)}$
Substituting $\tan^2(x) + 1 = \frac{1}{\cos^2(x)}$, we get:
$\frac{1 - (\frac{1}{\cos^2(x)} - 1)}{1 + (\frac{1}{\cos^2(x)} - 1)}$
Simplifying the Expression
Simplifying the expression, we get:
$\frac{\cos^2(x) - (1 - \cos^2(x))}{\cos^2(x) + (1 - \cos^2(x))}$
Combining like terms:
$\frac{2\cos^2(x) - 1}{1}$
Final Result
Therefore, the simplified expression is:
$\boxed{\cos(2x)}$
This result is obtained by recognizing that $2\cos^2(x) - 1 = \cos(2x)$, which is a well-known trigonometric identity.
In conclusion, we have successfully simplified the expression $\frac{1 - \tan^2(x)}{1 + \tan^2(x)}$ to $\cos(2x)$.
