${sin}^{2} (n + 1) A - {sin}^{2} (n) = sin (2 n + 1) A . sin A$

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Solution

${sin}^{2} (n + 1) A - {sin}^{2} n A = sin (2 n + 1) A . sin A$
${sin}^{2} a - {sin}^{2} b = sin (a - b) sin (a + b)$
$\Rightarrow sin (n + A + n A) sin (n A + A - n A)$
$sin (2 n + 1) A . sin A$

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