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Extrakt04 2022-06-30 Answered
Find cos 8 x if:
tan x tan 2 x = cot 2 x cot 3 x
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Answers (2)

frethi38
Answered 2022-07-01 Author has 16 answers
Let t := tan x and T := t 2 0 (assuming x R ), so
4 T 2 ( 3 T ) ( 1 T ) 2 ( 1 3 T ) = 1 ( T + 1 ) ( T 2 6 T + 1 ) = 0 T 2 6 T + 1 = 0 ,
and
cos 4 x = 2 ( 1 T 1 + T ) 2 1 = T 2 6 T + 1 ( 1 + T ) 2 = 0 cos 8 x = 1.
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Amber Quinn
Answered 2022-07-02 Author has 3 answers
Using the product-to-sum identity tan θ tan φ = cos ( θ φ ) cos ( θ + φ ) cos ( θ φ ) + cos ( θ + φ )
(1) 1 = tan 2 2 x cos 2 x cos 4 x cos 2 x + cos 4 x = 1 cos 2 2 x cos 2 2 x cos 2 x 2 cos 2 2 x + 1 cos 2 x + 2 cos 2 2 x 1
With t = cos 2 x
t 2 ( 2 t 2 + t 1 ) = ( 1 t 2 ) ( 2 t 2 + t + 1 ) 2 t 3 + 2 t 2 t 1 = 0 ( t + 1 ) ( 2 t 2 1 ) = 0 t { 1 , ± 2 2 }
The root t = 1 must be excluded because the denominator in (1) vanishes when cos 2 x = 1 , which leaves cos 2 x = ± 2 2
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