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Allyson Booker
Mathematics
22 October, 10:50
Prove that:
(tan20°) ^2 + (tan40°^) 2 + (tan80°) ^2=33
+4
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Armani Petersen
22 October, 10:57
0
Step-by-step explanation:
We have tanπ3=-tan2π3=tan4π3=3-√
Since tan (3x) = 3tanx-tan3x1-3tan2x
Thus tanπ/9,-tan2π/9, tan4π/9are solutions to the polynomial 3-√=3x-x31-3x2
On simplication, x3-33-√x2-3x+3-√=0. (1)
Since (tanπ9-tan2π9+tan4π9) 2
=tan2π9+tan22π9+tan24π9
+ 2 (-tanπ9tan2π9+tanπ8tan4π9-tan2π9tan4π9)
By sum and product pairs of roots in (1) above
(-33-√) 2=tan2π9+tan22π9+tan24π9+2*-3
∴tan2π9+tan22π9+tan24π9=33
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