Prove the following identity:
tan2 A + cot2 A + 2 = sec2 A cosec2 A
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cot2 A - cos2 A = cos2 A cot2 A
sec2 A + cosec2 A = sec2 A cosec2 A
sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A