Identitas Trigonometri: 1 + tan^2(x) / 1 - tan^2(x) = 1 / cos^2(x) - sin^2(x)
Introduction
In trigonometry, there are many important identities that relate to each other. One of the most interesting and useful identities is the one that states:
$\frac{1 + \tan^2(x)}{1 - \tan^2(x)} = \frac{1}{\cos^2(x) - \sin^2(x)}$
In this article, we will explore this identity and provide a proof for it.
Proof
To prove this identity, we can start by using the Pythagorean identity:
$\sin^2(x) + \cos^2(x) = 1$
Dividing both sides by $\cos^2(x)$, we get:
$\tan^2(x) + 1 = \frac{1}{\cos^2(x)}$
Subtracting 1 from both sides, we get:
$\tan^2(x) = \frac{1}{\cos^2(x)} - 1$
Rearranging the terms, we get:
$\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}$
Now, let's consider the left-hand side of the original identity:
$\frac{1 + \tan^2(x)}{1 - \tan^2(x)}$
Substituting the expression for $\tan^2(x)$, we get:
$\frac{1 + \frac{\sin^2(x)}{\cos^2(x)}}{1 - \frac{\sin^2(x)}{\cos^2(x)}}$
Simplifying the expression, we get:
$\frac{\cos^2(x) + \sin^2(x)}{\cos^2(x) - \sin^2(x)}$
Using the Pythagorean identity, we can simplify the numerator:
$\frac{1}{\cos^2(x) - \sin^2(x)}$
Which is the right-hand side of the original identity.
Conclusion
In this article, we have proven the identity:
$\frac{1 + \tan^2(x)}{1 - \tan^2(x)} = \frac{1}{\cos^2(x) - \sin^2(x)}$
This identity is a useful tool for simplifying trigonometric expressions and solving problems in trigonometry.
