How would you prove that tan &#x2061;<!-- ⁡ --> ( <mrow class="MJX-TeXAtom-ORD"> &

Extrakt04

Extrakt04

Answered question

2022-06-16

How would you prove that tan ( α ) + 2 tan ( 2 α ) + 4 tan ( 4 α ) + 8 cot ( 8 α ) = cot ( α ) ?

Answer & Explanation

Myla Pierce

Myla Pierce

Beginner2022-06-17Added 20 answers

You have:
cot ( 2 a ) = cot ( a ) 2 tan ( a ) 2
Thus:
cot a = ? tan a + 2 tan ( 2 a ) + 4 tan ( 4 a ) + 8 ( 1 2 cot ( 4 a ) 1 2 tan ( 4 a ) ) = ? tan a + 2 tan ( 2 a ) + 4 tan ( 4 a ) + 4 cot ( 4 a ) 4 tan ( 4 a ) = ? tan a + 2 tan ( 2 a ) + 4 ( 1 2 cot ( 2 a ) 1 2 tan ( 2 a ) ) = ? tan a + 2 tan ( 2 a ) + 2 cot ( 2 a ) 2 tan ( 2 a ) = ? tan a + 2 ( cot ( a ) 2 tan ( a ) 2 ) = ? tan a + cot a tan a cot a = cot a
Leland Morrow

Leland Morrow

Beginner2022-06-18Added 11 answers

After a little simplification,
t + 2 2 t 1 t 2 + 4 2 2 t 1 t 2 1 ( 2 t 1 t 2 ) 2 + 8 1 ( 2 2 t 1 t 2 1 ( 2 t 1 t 2 ) 2 ) 2 2 2 2 t 1 t 2 1 ( 2 t 1 t 2 ) 2 = 1 t .

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