I curious practical solution.(Step by step) ( 1 + tan &#x2061;<!-- ⁡ -->

Kash Brennan 2022-05-24 Answered
I curious practical solution.(Step by step)
( 1 + tan 5 ) ( 1 + tan 10 ) ( 1 + tan 15 ) ( 1 + tan 40 )
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Answers (2)

elladanzaez
Answered 2022-05-25 Author has 8 answers
If A + B = π 4 , Then ( 1 + tan A ) ( 1 + tan B ) = 2
So ( 1 + tan 5 0 ) ( 1 + tan 40 0 ) = 2
and ( 1 + tan 15 0 ) ( 1 + tan 30 0 ) = 2
So
( 1 + tan 5 0 ) ( 1 + tan 40 0 ) ( 1 + tan 10 0 ) ( 1 + tan 35 0 ) ( 1 + tan 15 0 ) ( 1 + tan 30 0 ) ( 1 + tan 20 0 ) ( 1 + tan 35 0 ) = 2 4 = 16
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Hayley Sanders
Answered 2022-05-26 Author has 1 answers
For example, observe that
1 = tan 45 = tan ( 5 + 40 ) = tan 5 + tan 40 1 tan 5 tan 40 .
Then we have tan 5 + tan 40 = 1 tan 5 tan 40 , and thus
( 1 + tan 5 ) ( 1 + tan 40 ) = 1 + tan 5 tan 40 + tan 5 + tan 40 = 1 + tan 5 tan 40 + 1 tan 5 tan 40 = 2.
Hence we conclude that
( 1 + tan 5 ) ( 1 + tan 10 ) ( 1 + tan 40 ) = 2 4 = 16.
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