Prove that tan 22 1/2 ° = √2 - 1
since tan θ/2 = (1-cosθ)/sinθ
tan π/8 = (1-1/√2)/(1/√2) = √2-1
To prove that tan(22.5°) = √2 - 1, we can use the half-angle formula for tangent:
tan(θ/2) = (1 - cosθ) / sinθ
Let's substitute θ = 45° into the formula:
tan(45°/2) = (1 - cos45°) / sin45°
Now, we need to find the values for cos45° and sin45°.
To do that, remember that in a right triangle, when one of the angles is 45°, the two legs are of equal length, and each leg is equal to the square root of 2 divided by 2.
cos45° = sin45° = √2/2
Now let's substitute those values into the formula:
tan(45°/2) = (1 - (√2/2)) / (√2/2)
Next, we need to simplify the expression:
Recall that (√2/2) can be written as (√2/2)^2 = 2/4 = 1/2.
tan(45°/2) = (1 - 1/2) / (1/2)
Now let's simplify the numerator:
1 - 1/2 = 2/2 - 1/2 = 1/2
Substituting this back into the expression:
tan(45°/2) = (1/2) / (1/2)
The denominator cancels out, leaving us with:
tan(45°/2) = 1
Therefore, we have proven that tan(22.5°) = 1.
Note: We obtained a different result, which means that the statement "tan(22.5°) = √2 - 1" is incorrect.