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simplifier les expressions suivantes : \( A=\cos 23^{\circ}+\cos ^{2} 75^{\circ}+\cos ^{2} 15^{\circ}-\sin 67^{\circ} \) \( B=\left(\sin ^{2} 65^{\circ}+\tan ^{2} 55^{\circ}+\sin ^{2} 25^{\circ}\right) \times \sin ^{2} 35^{\circ}-\tan 45^{\circ} \)

Ask by Allan Martin. in Morocco
Dec 24,2024

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Simplifiez les expressions suivantes : \( A = \cos 23^{\circ} + \cos^{2} 75^{\circ} + \cos^{2} 15^{\circ} - \sin 67^{\circ} \) \( B = \left( \sin^{2} 65^{\circ} + \tan^{2} 55^{\circ} + \sin^{2} 25^{\circ} \right) \times \sin^{2} 35^{\circ} - \tan 45^{\circ} \) La valeur simplifiée de \( A \) est 1. La valeur simplifiée de \( B \) est 0.

Solution

Calculez ou simplifiez l'expression \( A=\cos 23^{\circ}+\cos ^{2} 75^{\circ}+\cos ^{2} 15^{\circ}-\sin 67^{\circ} \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(23^{\circ}\right)+\cos^{2}\left(75^{\circ}\right)+\cos^{2}\left(15^{\circ}\right)-\sin\left(67^{\circ}\right)\) - step1: Subtract the numbers: \(0+\cos^{2}\left(75^{\circ}\right)+\cos^{2}\left(15^{\circ}\right)\) - step2: Remove 0: \(\cos^{2}\left(75^{\circ}\right)+\cos^{2}\left(15^{\circ}\right)\) - step3: Transform the expression: \(1\) Calculez ou simplifiez l'expression \( B=\left(\sin ^{2} 65^{\circ}+\tan ^{2} 55^{\circ}+\sin ^{2} 25^{\circ}\right) \times \sin ^{2} 35^{\circ}-\tan 45^{\circ} \). Calculate the value by following steps: - step0: Calculate: \(\left(\sin^{2}\left(65^{\circ}\right)+\tan^{2}\left(55^{\circ}\right)+\sin^{2}\left(25^{\circ}\right)\right)\sin^{2}\left(35^{\circ}\right)-\tan\left(45^{\circ}\right)\) - step1: Calculate: \(\left(\sin^{2}\left(65^{\circ}\right)+\tan^{2}\left(55^{\circ}\right)+\sin^{2}\left(25^{\circ}\right)\right)\sin^{2}\left(35^{\circ}\right)-1\) - step2: Calculate the trigonometric value: \(0\) La valeur simplifiée de l'expression \( A \) est 1. La valeur simplifiée de l'expression \( B \) est 0.

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Pour simplifier \( A \), on commence par utiliser quelques identités trigonométriques. On sait que \( \cos 75^{\circ} = \sin 15^{\circ} \), donc \( \cos^{2} 75^{\circ} = \sin^{2} 15^{\circ} \) et \( \cos 23^{\circ} \) reste tel quel. Également, \( \sin 67^{\circ} = \cos 23^{\circ} \), ce qui nous permet de réduire \( A \) : \[ A = \cos 23^{\circ} + \sin^{2} 15^{\circ} + \cos^{2} 15^{\circ} - \cos 23^{\circ} \] \( \sin^{2} 15^{\circ} + \cos^{2} 15^{\circ} = 1 \), donc : \[ A = 1 \] Pour le cas de \( B \), on commence par utiliser \( \tan 45^{\circ} = 1 \) pour réduire \( B \): \[ B = \left(\sin^{2} 65^{\circ} + \tan^{2} 55^{\circ} + \sin^{2} 25^{\circ}\right) \times \sin^{2} 35^{\circ} - 1 \] En utilisant \( \sin 65^{\circ} = \cos 25^{\circ} \) et \( \tan 55^{\circ} = \frac{\sin 55^{\circ}}{\cos 55^{\circ}} \) : \[ B = \left( \cos^{2} 25^{\circ} + \tan^{2} 55^{\circ} + \sin^{2} 25^{\circ} \right) \times \sin^{2} 35^{\circ} - 1 \] En remplaçant \( \cos^{2} 25^{\circ} + \sin^{2} 25^{\circ} = 1 \), on obtient : \[ B = \left( 1 + \tan^{2} 55^{\circ} \right) \sin^{2} 35^{\circ} - 1 \] Finalement, comme \( 1 + \tan^{2} 55^{\circ} = \sec^{2} 55^{\circ} \), on simplifie : \[ B = \sec^{2} 55^{\circ} \sin^{2} 35^{\circ} - 1 \] Pour des valeurs numériques plus concrètes, il serait intéressant de calculer ces résultats selon une calculatrice ou utiliser des tables trigonométriques, et vérifier que \( B \) reste positif à cause de l'interaction de sec et sin dans sa définition.

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