Question

Evaluate $\sum\frac{sin(A + B) sin(A - B)}{cos^{2} A . cos^{2} B}$; if none of cos A, cos B and cos C is zero.

Answer 1

Please see the attachment

Answer 2

Answer:

∑ Sin(A + B)Sin(A-B) / (cos²A.cos²B) = 0

Step-by-step explanation:

∑ Sin(A + B)Sin(A-B) / (cos²A.cos²B)

Sin(A + B)Sin(A-B)

using Sin(A + B) = SinACosB + CosASinB

& Sin(A - B) = SinACosB - CosASinB

= (SinACosB + CosASinB)(SinACosB - CosASinB)

= Sin²ACos²B - Cos²ASin²B

∑ Sin(A + B)Sin(A-B) / (cos²A.cos²B)

= ∑ (Sin²ACos²B - Cos²ASin²B)/ (cos²A.cos²B)

= ∑ (Sin²ACos²B(cos²A.cos²B) - Cos²ASin²B/(cos²A.cos²B))

= ∑ (Tan²A - Tan²B)

= Tan²A - Tan²B + Tan²B - Tan²C + Tan²C - Tan²A

= 0

∑ Sin(A + B)Sin(A-B) / (cos²A.cos²B) = 0