GABARITO: ALTERNATIVA C

𝑎. 𝑏 = (𝑠𝑒𝑛 𝑥 + 𝑠𝑒𝑛 𝑦). (cos 𝑥 + cos 𝑦) = 𝑠𝑒𝑛𝑥. 𝑐𝑜𝑠𝑥 + (𝑠𝑒𝑛𝑥. 𝑐𝑜𝑠𝑦 + 𝑠𝑒𝑛𝑦. 𝑐𝑜𝑠𝑥) + 𝑠𝑒𝑛𝑦. cos 𝑦= 𝑎. 𝑏 = (𝑠𝑒𝑛𝑥. 𝑐𝑜𝑠𝑦 + 𝑠𝑒𝑛𝑦. 𝑐𝑜𝑠𝑥) + (𝑠𝑒𝑛𝑥. 𝑐𝑜𝑠𝑥 + 𝑠𝑒𝑛𝑦. cos 𝑦) = 𝑠𝑒𝑛 (𝑥 + 𝑦) + (𝑠𝑒𝑛𝑥. 𝑐𝑜𝑠𝑥 + 𝑠𝑒𝑛𝑦. cos 𝑦)

Para (𝑠𝑒𝑛𝑥. 𝑐𝑜𝑠𝑥 + 𝑠𝑒𝑛𝑦. cos 𝑦)

(𝑠𝑒𝑛 2𝑥 + sen 2𝑦)

Para: 2𝑥 = 𝑝 + 𝑞 𝑒 2𝑦 = 𝑝 − 𝑞 → 𝑝 = 𝑥 + 𝑦 𝑒 𝑞 = 𝑥 − 𝑦

Logo:

(𝑠𝑒𝑛 2𝑥 + sen 2𝑦) =

𝑠𝑒𝑛 (𝑝 + 𝑞) + 𝑠𝑒𝑛(𝑝 − 𝑞) = 𝑠𝑒𝑛 𝑝 . cos 𝑞 + 𝑠𝑒𝑛 𝑞 . cos 𝑝 + 𝑠𝑒𝑛 𝑝 . cos 𝑞 − 𝑠𝑒𝑛 𝑞 . cos 𝑝 = 2𝑠𝑒𝑛 𝑝. cos q

2. 𝑠𝑒𝑛(𝑥 + 𝑦). cos (𝑥 − 𝑦)

Então: (𝑠𝑒𝑛𝑥. 𝑐𝑜𝑠𝑥 + 𝑠𝑒𝑛𝑦. cos 𝑦)

(𝑠𝑒𝑛 2𝑥 + sem 2𝑦) = 𝑠𝑒𝑛(𝑥 + 𝑦). cos(𝑥 − 𝑦)

𝑎. 𝑏 = 𝑠𝑒𝑛 (𝑥 + 𝑦) + (𝑠𝑒𝑛(𝑥 + 𝑦). cos(𝑥 − 𝑦)) = 𝑠𝑒𝑛(𝑥 + 𝑦)(1 + cos(𝑥 − 𝑦))

𝑎² + 𝑏² = 𝑠𝑒𝑛 2𝑥 + 2. 𝑠𝑒𝑛 𝑥. 𝑠𝑒𝑛 𝑦 + 𝑠𝑒𝑛²𝑦 + 𝑐𝑜𝑠²𝑥 + 2. cos 𝑥 . cos 𝑦 + 𝑐𝑜𝑠²𝑦 = 2 + 2. cos 𝑥 . cos 𝑦 + 2. 𝑠𝑒𝑛 𝑥. 𝑠𝑒𝑛 y 𝑎² + 𝑏² = 2(1 + cos(𝑥 − 𝑦))

Gabarito: C