P.T ${sin}^{2} A - {sin}^{2} B = sin (A + B) - sin (A - B)$

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Solution

$sin (A + B) = sin A cos B + cos A sin B$
Now $sin (- B) = - sin (B)$ and $cos (- B) = cos B$ , so
$sin (A - B) = sin A cos B - cos (A) sin B$
So :-
$sin (A + B) . sin (A - B)$
$= (sin A cos B + cos A sin B) (sin A cos B - cos A sin B)$
$= (sin A cos B)^{2} - (cos A sin B)^{2}$
$= {sin}^{2} A {cos}^{2} B - {cos}^{2} A {sin}^{2} B$
$= {sin}^{2} A (1 - {sin}^{2} B) - {sin}^{2} B (1 - {sin}^{2} A)$
$= {sin}^{2} A - {sin}^{2} B$

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