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Conditional Identities

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$\frac{t a n 5 A - t a n 3 A}{t a n 5 A + t a n 3 A} = \frac{s i n 2 A}{s i n 8 A} .$

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Solution

$\frac{tan 5 A - tan 3 A}{tan 5 A + tan 3 A} = \frac{\frac{sin 5 A}{cos 5 A} - \frac{sin 3 A}{cos 3 A}}{\frac{sin 5 A}{cos 5 A} + \frac{sin 3 A}{cos 3 A}}$
$= \frac{sin 5 A cos 3 A - sin 3 A cos 5 A}{(cos 5 A cos 3 A) \frac{(sin 5 A cos 3 A + sin 3 A cos 5 A)}{cos 5 A cos 3 A}}$
$= \frac{sin (5 A - 3 A)}{sin (3 A + 5 A)} = \frac{sin 2 A}{sin 8 A}$

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