What is the exact value of tangent of open paren 67.5 degrees close paren ?

Use the half-angle identity: tan(67.5°) = cot(22.5°). Since tan(22.5°) = √2 − 1, its reciprocal is 1/(√2 − 1) = √2 + 1. Choice D equals this value because √((2 + √2)/(2 − √2)) = √(3 + 2√2) = √2 + 1.

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