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Use the identity tanA = sin/cosA
Tan²A - tan²B = sin²A - sin²B/ cos²A cos²B
LHS = tan²A - tan²B
= sin²A/ cos²A - sin²B/ cos²B
= sin²A cos²B - sin²B cos²A/ cos²A cos²B
= sin²A (1 - sin²B) - sin²B (1 - sin²A) / cos²A cos²B
= sin²A - sin²B sin²A - sin²B + sin²B / cos²A cos²B
= sin²A - sin²B / cos²A cos²B = RHS
Hope this helps!