Question 1210351: Find the exact value of
tan⁸40° - 36tan⁶40° + 126tan⁴40 - 84tan²40 Found 2 solutions by ikleyn, Edwin McCravy:Answer by ikleyn(52754) (Show Source):
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Find the exact value of
tan⁸(40°) - 36tan⁶(40°) + 126tan⁴(40°) - 84tan²(40°)
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Consider complex number z = cos(40°) + i*sin(40°) = cis(40°).
Notice that 9*40° = 360°. Therefore, = cis(9*40°) = cis(360°) = 1 + i*0 = 1.
It is the same as to say
= 1. (1)
Apply the Newton binomial decomposition formula. Keep only the terms with odd degrees of 'i'
and equate their sum to zero (since the imaginary part in the right side of formula (1) is zero)
+ + + + = 0.
Divide both sides by . Take into account that = -1, = 1, = -1, = 1. You will get
- + - + = 0.
Divide both sides by tan(40°). You will get
- + - = -9.
ANSWER. - + - = -9.
