Review the proof of cos A-B
The Correct Answer and Explanation is:
Here is the correct choice and a detailed explanation of the reasoning.
Correct Answer:
The correct option to complete step 4 of the proof is 2 and 1.
Explanation
The purpose of Step 4 is to simplify the expanded equation from Step 3 by applying the fundamental Pythagorean identity, which states that for any angle θ, cos²θ + sin²θ = 1. Let’s break down how this is applied to both sides of the equation.
Step 3:
The equation after expanding the squares in Step 2 is:
cos²A – 2cosAcosB + cos²B + sin²A – 2sinAsinB + sin²B = cos²(A – B) – 2cos(A – B) + 1 + sin²(A – B)
Deriving Step 4:
- Simplify the Left-Hand Side (LHS):
We can regroup the terms on the LHS of the equation from Step 3 to pair up the squared sine and cosine terms:
(cos²A + sin²A) + (cos²B + sin²B) – 2cosAcosB – 2sinAsinB
Applying the Pythagorean identity to (cos²A + sin²A) gives 1.
Applying the Pythagorean identity to (cos²B + sin²B) also gives 1.
Substituting these values back, the LHS simplifies to:
1 + 1 – 2cosAcosB – 2sinAsinB
= 2 – 2cosAcosB – 2sinAsinB
Therefore, the first blank in Step 4 is 2. - Simplify the Right-Hand Side (RHS):
Similarly, we regroup the terms on the RHS of the equation from Step 3:
(cos²(A – B) + sin²(A – B)) – 2cos(A – B) + 1
Applying the Pythagorean identity to (cos²(A – B) + sin²(A – B)) gives 1.
Substituting this value back, the RHS becomes:
1 – 2cos(A – B) + 1
Therefore, the second blank in Step 4 is 1.
Completed Step 4:
By filling in the blanks with our derived values, the complete Step 4 is:
2 – 2cosAcosB – 2sinAsinB = 1 – 2cos(A – B) + 1
This equation simplifies further to 2 – 2cosAcosB – 2sinAsinB = 2 – 2cos(A – B). Subtracting 2 from both sides leads directly to Step 5, confirming that our choice is the correct logical progression in the proof.
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The Correct Answer and Explanation is: