In-ABC-cot-A-2-cot-B-2-cot-C-2-Q-Prove-that-cyc-sin-A-2-n-cyc-cos-A-2-n-Q-n-N-

Question Number 176128 by Shrinava last updated on 13/Sep/22

In △ABC , cot (A/2) , cot (B/2) , cot (C/2) ∈ Q Prove that: (Π_(cyc) sin (A/2))^n + (Π_(cyc) cos (A/2))^n ∈ Q , ∀n∈N

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\mathrm{cot}\:\frac{\mathrm{A}}{\mathrm{2}}\:,\:\mathrm{cot}\:\frac{\mathrm{B}}{\mathrm{2}}\:,\:\mathrm{cot}\:\frac{\mathrm{C}}{\mathrm{2}}\:\in\:\mathrm{Q} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\left(\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} +\:\left(\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\mathrm{cos}\:\frac{\mathrm{A}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} \in\:\mathrm{Q}\:,\:\forall\mathrm{n}\in\mathbb{N} \\ $$

Commented by Rasheed.Sindhi last updated on 13/Sep/22

Shrinava sir, are you an old member of the forum? I mean have you a different ID in the past?

$$\mathrm{Shrinava}\:\mathrm{sir},\:{are}\:{you}\:{an}\:{old}\:{member} \\ $$$${of}\:{the}\:{forum}?\:{I}\:{mean}\:{have}\:{you}\:{a} \\ $$$${different}\:{ID}\:{in}\:{the}\:{past}? \\ $$

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