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My reasoning is this:

The hypotenuse of a 1,1,√2 right triangle is at a 45° angle to its base.

Halving that angle should require that the base leg be doubled if the height leg is kept at 1, giving the new triangle lengths of 1,2,√5

The cosine is adjacent/hypotenuse which would be 2/√5, yet calculators give a different answer. Why is this?