https://file.aiquickdraw.com/mathbot/file/1724482789098cqj3hf4v.jpg

Let's break down and solve the trigonometric identities exercise from the image.

### 1. Prove the identity:
\[
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\]

This is a standard trigonometric identity for the sine of a sum of two angles. It states:

\[
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\]

This identity is derived from the unit circle or using Euler's formulas for sine and cosine. Therefore, this identity is already proven and is commonly used in trigonometry.

### 2. Show that \(\sin^2 x + \cos^2 x = 1\) and then calculate \(\sin^2 \left(\frac{\pi}{3}\right) + \cos^2 \left(\frac{\pi}{3}\right)\).

This is another fundamental trigonometric identity, known as the Pythagorean identity:

\[
\sin^2 x + \cos^2 x = 1
\]

Let's compute \(\sin^2 \left(\frac{\pi}{3}\right) + \cos^2 \left(\frac{\pi}{3}\right)\):

- \(\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)
- \(\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}\)

Now, square these values and add them together:

\[
\sin^2 \left(\frac{\pi}{3}\right) + \cos^2 \left(\frac{\pi}{3}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1
\]

Thus, the identity holds true for \(x = \frac{\pi}{3}\).

### 3. Calculate \(\sin^2 x + \cos^2 x\) using \(x = \frac{\pi}{3}\) and verify that the identity \(\sin^2 x + \cos^2 x = 1\) holds.

This part essentially repeats what we just calculated in step 2. For \(x = \frac{\pi}{3}\), we already verified that:

\[
\sin^2 \left(\frac{\pi}{3}\right) + \cos^2 \left(\frac{\pi}{3}\right) = 1
\]

### 4. Simplify the expression \(\cos \left(\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{3}\right)\) and \(\cos^2 \left(\frac{\pi}{4}\right) - \sin^2 \left(\frac{\pi}{4}\right)\).

For \(\cos \left(\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{3}\right)\):

- \(\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}\)
- \(\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)

So:

\[
\cos \left(\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{3}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2}
\]

For \(\cos^2 \left(\frac{\pi}{4}\right) - \sin^2 \left(\frac{\pi}{4}\right)\):

- \(\cos \left(\frac{\pi}{4}\right) = \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)

Therefore:

\[
\cos^2 \left(\frac{\pi}{4}\right) - \sin^2 \left(\frac{\pi}{4}\right) = \left(\frac{\sqrt{2}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} - \frac{2}{4} = 0
\]

### Summary:
1. The first identity is a fundamental trigonometric identity for the sine of a sum.
2. The second part verifies the Pythagorean identity using a specific angle.
3. The calculations confirm the identity \(\sin^2 x + \cos^2 x = 1\) for \(x = \frac{\pi}{3}\).
4. The simplification of trigonometric expressions gives specific values depending on the angle.

Would you like further details on any of these problems, or have any other questions? 

Here are some related questions you might consider:
1. What is the cosine of the sum of two angles?
2. How do you prove the identity \(\cos^2 \theta + \sin^2 \theta = 1\) using the unit circle?
3. What is the relationship between \(\sin(x)\) and \(\cos(x)\) in terms of their derivatives?
4. How do you simplify trigonometric expressions involving double angles?
5. What are some real-world applications of trigonometric identities?

**Tip:** Always remember that understanding the derivation of these identities helps reinforce their application and ensures accuracy in problem-solving.

Explore and solve trigonometric identities exercises step-by-step, including proving identities and calculating specific values like sin^2(pi/3) + cos^2(pi/3). Learn fundamental trigonometric concepts and theorems. Suitable for high school students.

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