https://file.aiquickdraw.com/mathbot/file/1725561420095ht2r0otx.jpg

From the uploaded image, the task involves completing three trigonometric identities:

1. **Double angle formula for sine:**
   \[
   \sin(2x) = 2 \sin(x) \cos(x)
   \]

2. **Half angle formula for sine:**
   \[
   \sin^2\left(\frac{x}{2}\right) = \frac{1 - \cos(x)}{2}
   \]

3. **Half angle formula for cosine:**
   \[
   \cos^2\left(\frac{x}{2}\right) = \frac{1 + \cos(x)}{2}
   \]

Would you like more details or explanations on these formulas?

Here are 5 follow-up questions to further explore:

1. How do you derive the double angle formula for sine?
2. What is the relationship between the sine and cosine double angle formulas?
3. How is the half-angle formula for sine used in integration?
4. Can you explain the geometric meaning of the half-angle identities?
5. How do these trigonometric identities simplify solving trigonometric equations?

**Tip**: The double and half-angle identities are useful tools for solving trigonometric equations and simplifying integrals or derivatives involving trigonometric functions.

Fill in the following trigonometric identities: (a) Double angle formula for sine: sin(2x) = ? (b) Half angle formula for sine: sin²(x/2) = ? (c) Half angle formula for cosine: cos²(x/2) = ?

Learn how to fill in the double and half-angle formulas for sine and cosine. This problem includes the key trigonometric identities: sin(2x), sin²(x/2), and cos²(x/2), using fundamental trigonometry concepts. Suitable for high school students (Grades 10-12).

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