The identity tan(45+A) tan(45-A) = 1 can be proven using the tangent addition and subtraction formulas. Specifically, the tangent addition formula states that tan(a+b) = (tan a + tan b) / (1 - tan a tan b), and the subtraction formula states that tan(a-b) = (tan a - tan b) / (1 + tan a tan b). By substituting A for the variable and recognizing that tan(45 degrees) equals 1, the identity holds true without the need for numerical calculations.
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