When you're learning about trigonometric identities, it can be easy to make some mistakes. Here are some common ones to watch out for:

1. Ignoring the Domain

Different trigonometric functions have certain areas where they work. For example, the tangent function, which is written as $\tan(x) = \frac{\sin(x)}{\cos(x)}$, doesn’t have a value where $\cos(x) = 0$. This happens at $x = \frac{\pi}{2} + k\pi$, where $k$ can be any whole number. So, always check for these limits!

2. Misusing Pythagorean Identities

Pythagorean identities can be tricky. For example, $1 + \tan^2(x) = \sec^2(x)$ is an important one, but make sure you know when to use it. If you have $\tan(x)$, you can find $\sec(x)$, but don't mix them up with their opposite values when you simplify.

3. Thinking You Can Rearrange Everything

Not every operation you do with trigonometric functions can be changed around without thought. For instance, $sin(a + b)$ is not the same as $sin(a) + sin(b)$.

4. Forgetting About Reciprocal and Quotient Identities

Always keep your reciprocal identities in mind, like $\csc(x) = \frac{1}{\sin(x)}$. Not remembering these can lead to mistakes when you're trying to simplify things.

5. Not Practicing Enough

Trigonometric identities can be hard! The best way to understand them is to practice. Try solving problems to make things clearer. For example, work on simplifying $\frac{\sin^2(x)}{1 - \cos(x)}$ using what you’ve learned about identities.

By keeping these common mistakes in mind, you'll get better at trigonometric identities before you know it!