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$$\tan A + \sin A = p; \tan A - \sin A = q \Rightarrow p^2 - q^2 = $$
$$2(\tan^2 A + \sin^2 A)$$
$$2(\tan^2 A - \sin^2 A)$$
$$4 \tan^2 A \sin^2 A$$
$$4 \tan A \sin A$$
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