To solve this, we use the Sine Rule, which states:
sinAa=sinBb=sinCc=k
This implies sinA=ak,sinB=bk, and sinC=ck. Also, in any triangle, A+B+C=180∘, so A+B=180∘−C.
Step 1: Simplify the denominator
Using the property sin(180∘−θ)=sinθ:
sin(A+B)=sin(180∘−C)=sinC
Step 2: Use the Sine Rule to express the ratio
The given expression becomes:
sinCsin(A−B)=sinCsinAcosB−cosAsinB
Substituting sinA=ak,sinB=bk, and sinC=ck:
=ckakcosB−bkcosA=cacosB−bcosA
Step 3: Apply the Projection Formula
From the projection formula, we know:
a=bcosC+ccosB⟹ccosB=a−bcosC
b=acosC+ccosA⟹ccosA=b−acosC
Multiply the numerator and denominator by c:
c2c(acosB)−c(bcosA)=c2a(ccosB)−b(ccosA)
Substitute the projection identities:
=c2a(a−bcosC)−b(b−acosC)
=c2a2−abcosC−b2+abcosC
=c2a2−b2
Final Answer:
The expression is equal to c2a2−b2. The correct option is (c).
