\lim _{x\to \pi }\left(\frac{1+\cos \left(x\right)}{\left(x-\pi \right)^2}\right) only using trigonometry identity

To solve the limit \(\lim _{x\to \pi }\left(\frac{1+\cos \left(x\right)}{\left(x-\pi \right)^2}\right)\) using trigonometric identities, let's proceed step by step.

### Step 1: Express \(\cos(x)\) in a different form
We know from the trigonometric identity:

\[
\cos(x) = 1 - 2\sin^2\left(\frac{x}{2}\right)
\]

However, to express \(\cos(x)\) near \(x = \pi\), we can use the identity:

\[
\cos(x) = -\cos(\pi - x)
\]

So:

\[
\cos(x) = -\cos(x - \pi)
\]

### Step 2: Substitute the identity into the limit expression
Now, substitute \(\cos(x) = -\cos(x - \pi)\) into the limit expression:

\[
\lim _{x\to \pi }\left(\frac{1+\cos \left(x\right)}{\left(x-\pi \right)^2}\right) = \lim _{x\to \pi }\left(\frac{1 - \cos(x - \pi)}{(x - \pi)^2}\right)
\]

### Step 3: Use the small angle approximation
For \(x\) close to \(\pi\), \(x - \pi\) is small. We can use the approximation:

\[
\cos(y) \approx 1 - \frac{y^2}{2} \quad \text{for small } y
\]

Thus, \(\cos(x - \pi) \approx 1 - \frac{(x - \pi)^2}{2}\).

### Step 4: Substitute the approximation back into the limit
Substitute this into the expression:

\[
\lim _{x\to \pi }\left(\frac{1 - \left(1 - \frac{(x - \pi)^2}{2}\right)}{(x - \pi)^2}\right)
\]

Simplify the expression:

\[
\lim _{x\to \pi }\left(\frac{\frac{(x - \pi)^2}{2}}{(x - \pi)^2}\right)
\]

\[
\lim _{x\to \pi }\left(\frac{1}{2}\right) = \frac{1}{2}
\]

### Final Answer:
\[
\lim _{x\to \pi }\left(\frac{1+\cos \left(x\right)}{\left(x-\pi \right)^2}\right) = \frac{1}{2}
\]

Would you like more details or have any questions?

Here are five related questions you might find interesting:
1. How do small angle approximations work in other trigonometric functions?
2. What are other methods to solve limits involving trigonometric functions?
3. How does L'Hôpital's Rule apply to this limit?
4. How can Taylor series be used to solve limits near critical points?
5. What is the significance of trigonometric identities in calculus?

**Tip:** Always consider whether small angle approximations or trigonometric identities can simplify complex trigonometric limits.

Learn how to solve the limit lim_(x -> pi) ((1 + cos(x)) / (x - pi)^2) using trigonometric identities and small angle approximations. This problem involves advanced concepts in trigonometry and limits. Suitable for advanced high school students and college-level calculus enthusiasts.

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