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Question -

a2 sin(B – C) = (b2 – c2) sin A



Answer -

By using the sine rule we know,

So, c = k sin C

Similarly, a = k sin A

And b = k sin B

We know,

Now let us consider RHS:

(b2 – c2) sin A …

Substituting the values of b and c in the aboveequation, we get

(b2 – c2) sin A = [(k sinB)2 – ( k sin C)2] sin A

= k(sin2 B – sin2 C)sin A………. (i)

We know,

Sin2 B – sin2 C = sin(B + C) sin (B – C),

Substituting the above values in equation (i), we get

 = k(sin (B + C) sin (B – C))sin A [since, π = A + B + C  B + C = π –A]

 = k(sin (π –A) sin (B – C))sin A

= k(sin (A) sin (B – C)) sin A[since, sin (π – θ) = sin θ]

Rearranging the above equation we get

= (k sin (A))( sin (B – C)) (k sin A)

From sine rule, a = k sin A, so the above equationbecomes,

= a2 sin (B – C)

= RHS

Hence proved.

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