Show that tan &#x2061;<!-- ⁡ --> 72 &#x2218;<!-- ∘ --> </msup> &#x2212;<!-- - -

2d3vljtq

2d3vljtq

Answered question

2022-07-02

Show that
tan 72 cot 36 cot 54 = 3 tan 18

Answer & Explanation

billyfcash5n

billyfcash5n

Beginner2022-07-03Added 17 answers

A little generalization:
Replacing 18 with t,
1 tan t 3 tan t = 4 cos 2 t 3 sin t cos t = 2 cos 3 t cos t sin 2 t = 2 cos 3 t ( 2 cos 2 t ) cos t sin 4 t = 2 { cos ( 3 t 2 t ) + cos ( 3 t + 2 t ) } cos t sin 4 t
if cos t 0 and cos 5 t = 0 5 t = ( 2 n + 1 ) 90
t = ( 2 n + 1 ) 18 where n 2 ( mod 5 )
Now tan 2 t + 1 tan 2 t = 2 sin 4 t
Here n = 0
uplakanimkk

uplakanimkk

Beginner2022-07-04Added 6 answers

cot 36 + cot 54 = 2 sin 72         ( 1 )
tan 72 tan 18 = sin ( 72 18 ) cos 18 sin 18 = 2 sin 54 sin 36 = 2 cot 36         ( 2 )
cot 36 tan 18 = cot 36 cot 72 = sin ( 72 36 ) sin 36 sin 72 = 1 sin 72         ( 3 )
tan 72 cot 36 cot 54 3 tan 18
= ( tan 72 tan 18 ) ( cot 36 + cot 54 ) 2 tan 18
= 2 ( cot 36 tan 18 ) 2 sin 72 ( using  ( 2 ) , ( 1 ) )
Now use (3)

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