Simplify the expression (sin A - sin B) (cos A + cos B). Solution: Step 1. Multiply the expression by distributive property. sin A cos A + sin A cos B - sin B cos A - sin B cos B Step 2. Eliminate the second and third terms. sin A cos A - sin B cos B Step 3. Use Product-to-Sum identity. (1/2) [sin (A + A) + sin (A - A)] - (1/2) [sin (B + B) + sin (B - B)] Step 4. Simplify. (1/2) sin (2A) - (1/2) sin (2B) Is the suggested solution correct? Explain with reasons.
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The distributive property has been applied correctly. Step 2 is also correct. Show more…
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Simplify the expression $(\sin A-\sin B)(\cos A+\cos B)$ Solution: Multiply the expressions using the distributive property. $$\sin A \cos A+\sin A \cos B-\sin B \cos A-\sin B \cos B$$ Cancel the second and third terms. $$\sin A \cos A-\sin B \cos B$$ Use the product-to-sum identity. $$\underbrace{\sin A \cos A}_{\frac{1}{2}[\sin (A+A)+\sin (A-A)]}-\frac{\sin B \cos B}{\frac{1}{2}[\sin (B+B)+\sin (B-B)]}$$ Simplify. $=\frac{1}{2} \sin (2 A)-\frac{1}{2} \sin (2 B)$ This is incorrect. What mistake was made?
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Explain the mistake that is made. Simplify the expression $(\sin A-\sin B)(\cos A+\cos B)$ Solution: Multiply the expressions using the distributive property. $$\sin A \cos A+\sin A \cos B-\sin B \cos A-\sin B \cos B$$ Cancel the second and third terms. $$\sin A \cos A-\sin B \cos B$$ Use the product-to-sum identity. $$\underbrace{\sin A \cos A}_{\frac{1}{2}[\sin (A+A)+\sin (A-A)]}-\underbrace{\sin B \cos B}_{\frac{1}{2}[\sin (B+B)+\sin (B-B)]}$$ Simplify. $\quad=\frac{1}{2} \sin (2 A)-\frac{1}{2} \sin (2 B)$ This is incorrect. What mistake was made?
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