In any - ABC- prove that a2 - b2c2 - sinA-B sin A - B-

Consider the given equation. sin( A B )sin( A+B )=a^2 b^2c^2 L.H.S. , sin( A B )sin( A+B #10;)=sin( A B )sin( A+B )×sin #10;( A+B )sin( A+B ) ...

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Byju's Answer

Standard VII

Mathematics

Equal Angles Subtend Equal Sides

In any Δ AB...

Question

In any $Δ A B C$ , prove that $\frac{a^{2} - b^{2}}{c^{2}} = \frac{sin (A - B)}{sin (A + B)}$ .

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Solution

Consider the given equation.
$\frac{sin (A - B)}{sin (A + B)} = \frac{a^{2} - b^{2}}{c^{2}}$
$L . H . S .$ ,
$\frac{sin (A - B)}{sin (A + B)} = \frac{sin (A - B)}{sin (A + B)} \times \frac{sin (A + B)}{sin (A + B)}$
$= \frac{sin (A - B) sin (A + B)}{{sin}^{2} (A + B)} = \frac{{sin}^{2} A - {sin}^{2} B}{{sin}^{2} (π - c)}$
Put, $sin A = a k, sin B = b k$ and $sin C = c k$ .
$\frac{{sin}^{2} A - {sin}^{2} B}{{sin}^{2} (π - c)} = \frac{a^{2} k^{2} - b^{2} k^{2}}{c^{2} k^{2}}$
$= \frac{k^{2} (a^{2} - b^{2})}{c^{2}} = \frac{a^{2} - b^{2}}{c^{2}} = R H S$
Hence, proved.

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In any ΔABC, prove that a2−b2c2=sin(A−B)sin(A+B).

Consider the given equation.sin(A−B)sin(A+B)=a2−b2c2L.H.S.,sin(A−B)sin(A+B)=sin(A−B)sin(A+B)×sin(A+B)sin(A+B)=sin(A−B)sin(A+B)sin2(A+B)=sin2A−sin2Bsin2(π−c)Put, sinA=ak,sinB=bk and sinC=ck.sin2A−sin2Bsin2(π−c)=a2k2−b2k2c2k2=k2(a2−b2)c2=a2−b2c2=RHSHence, proved.

In any $Δ A B C$ , prove that $\frac{a^{2} - b^{2}}{c^{2}} = \frac{sin (A - B)}{sin (A + B)}$ .