[attach]87027[/attach]

$ begin {aligned} sin^2( frac {α}{2}+ frac {π}{4}) &= [sin( frac {α}{2}+ frac {π}{4})]^2 &= [sin( frac {α}{2})cos( frac {π}{4}) + sin( frac {π}{4})cos( frac {α}{2})]^2 &= [ frac {√2}{2} (sin( frac {α}{2})+cos( frac {α}{2}))]^2 &= frac {1}{2}[+/- sqrt { frac {1-cos(α)}{2}} +/- sqrt { frac {1+cos(α)}{2}}]^2 &= frac {1}{2}( frac {1-cos(α)}{2} + frac {1+cos(α)}{2} + 2 sqrt { frac {1-cos^2(α)}{4}}) &= frac {1}{2} ( frac {2}{2} + sin(α)) &= frac {1}{2}(1+sin(α)) end {aligned}

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serena_trevisan

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21/08/2024 22:01

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$ \begin{aligned} sin^2(\frac{α}{2}+\frac{π}{4}) &= [sin(\frac{α}{2}+\frac{π}{4})]^2 \\ &= [sin(\frac{α}{2})cos(\frac{π}{4}) + sin(\frac{π}{4})cos(\frac{α}{2})]^2 \\ &= [\frac{√2}{2} (sin(\frac{α}{2})+cos(\frac{α}{2}))]^2 \\&= \frac{1}{2}[\pm \sqrt{\frac {1-cos(α)}{2}} \pm \sqrt{\frac {1+cos(α)}{2}}]^2 \\ &= \frac{1}{2}(\frac {1-cos(α)}{2} + \frac {1+cos(α)}{2} + 2\sqrt{\frac{1-cos^2(α)}{4}}) \\ &= \frac{1}{2} (\frac{2}{2} + sin(α)) \\ &= \frac{1}{2}(1+sin(α)) \end{aligned}$