Question

a expressão sen ( x/2 + cos x/2 ) ² é equivalente á?

Answer 1

$\mathrm{\bigg(\sin{\dfrac{x}{2}}+\cos{\dfrac{x}{2}}\bigg)^2=\bigg(\sqrt{\dfrac{1-\cos{x}}{2}}+\sqrt{\dfrac{1+\cos{x}}{2}}\bigg)^2=}\\\\\\ \mathrm{=\dfrac{1-\cos{x}}{2}+2\sqrt{\dfrac{1-\cos{x}}{2}}\sqrt{\dfrac{1+\cos{x}}{2}}+\dfrac{1+\cos{x}}{2}=}\\\\\\ \mathrm{=\dfrac{1-\cos{x}+1+\cos{x}}{2}+2\dfrac{\sqrt{(1-\cos{x})(1+\cos{x})}}{2}=}\\\\\\ \mathrm{=\dfrac{2}{2}+\sqrt{1-\cos^2{x}}=1+\sqrt{\sin^2{x}}=\boxed{\mathbf{1+\sin{x}}}}$