Having trouble proving 4 tan <mrow class="MJX-TeXAtom-ORD"> &#x2212;<!-- -

Cory Patrick

Cory Patrick

Answered question

2022-06-27

Having trouble proving 4 tan 1 1 5 tan 1 1 70 + tan 1 1 99 = π 4

Answer & Explanation

Amy Daniels

Amy Daniels

Beginner2022-06-28Added 20 answers

Set tan α = 1 5 . Use the duplication formula tan 2 A = 2 tan A 1 tan 2 A twice to obtain tan 4 α = 120 119 . Hence, tan 4 α 1 = 1 119 and tan ( 4 α π 4 ) = 1 119 ( 1 + 120 119 ) = 1 239 from the fact that tan ( A B ) = tan A tan B 1 + tan A tan B . Hence, arctan 1 239 = 4 α π 4
On the other hand, arctan 1 70 arctan 1 99 can be calculated from the formula you gave at first, and it gives arctan 1 239
Hence, arctan 1 70 arctan 1 99 = 4 arctan 1 5 π 4 . Rearrange to get the desired result.

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