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Sin(a + 30°) - sin(a - 30°) = cos(a). How can I prove that?

A. By using the double angle formula for sine
B. By using the Pythagorean identity for sine
C. By using the sum and difference formula for sine
D. By using the sum and difference formula for cosine

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To prove the equation sin(a + 30°) - sin(a - 30°) = cos(a), we use the sum and difference formulas for sine. After applying these formulas, it simplifies to cos(a), confirming that option C is the correct method to use.

To prove that sin(a + 30°) - sin(a - 30°) = cos(a), we can use the sum and difference formulas for sine, which are formula number 10 in the list you provided. Applying the formulas, we have:


  • sin(a + 30°) = sin(a)cos(30°) + cos(a)sin(30°)

  • sin(a - 30°) = sin(a)cos(30°) - cos(a)sin(30°)

Subtracting the second equation from the first, we get:

sin(a + 30°) - sin(a - 30°) = (sin(a)cos(30°) + cos(a)sin(30°)) - (sin(a)cos(30°) - cos(a)sin(30°))

This simplifies to:

2cos(a)sin(30°) = cos(a)(2sin(30°)) = cos(a)(2 * 1/2) = cos(a)

Thus, proving the given equation. Therefore, the correct answer is C. By using the sum and difference formula for sine.

User ChrisWay
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