Find the value of tan8π.
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Step-by-Step Solution
Step 1: Use the half-angle identity for tangent
To find the value of tan8π, we can use the half-angle identity for tangent. In this case, we can set 2θ=8π, which means θ=4π. This identity allows us to express the tangent of a half-angle in terms of trigonometric functions of the full angle.
Step 2: Substitute the value of θ
Now, we substitute θ=4π into the half-angle identity. This gives us an expression for tan8π in terms of cos4π and sin4π.
Step 3: Evaluate sin4π and cos4π
We know the exact values for the sine and cosine of 4π (or 45∘). Both sin4π and cos4π are equal to 21.
Step 4: Substitute the values and simplify
Substitute the values of sin4π and cos4π into the expression. Then, simplify the complex fraction by multiplying the numerator and denominator by 2. This leads to the final simplified value for tan8π.