If sin 2A = 2 sin A cos A, what is the double angle formula for cosine?

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Q: If sin 2A = 2 sin A cos A, what is the double angle formula for cosine?

Solution: The double angle formula for cosine is cos 2A = cos²A - sin²A.

Steps: 7

Step 1: Understand that sin 2A is a trigonometric identity that can be expressed in terms of sin A and cos A.
Step 2: Recall the double angle formula for sine, which states that sin 2A = 2 sin A cos A.
Step 3: To find the double angle formula for cosine, we need to use the Pythagorean identity, which states that sin²A + cos²A = 1.
Step 4: Rearrange the Pythagorean identity to express sin²A in terms of cos²A: sin²A = 1 - cos²A.
Step 5: Substitute sin²A into the expression for cos 2A. We can express cos 2A in two ways: cos 2A = cos²A - sin²A or cos 2A = cos²A - (1 - cos²A).
Step 6: Simplify the second expression: cos 2A = cos²A - 1 + cos²A = 2cos²A - 1.
Step 7: Therefore, we have two forms of the double angle formula for cosine: cos 2A = cos²A - sin²A and cos 2A = 2cos²A - 1.