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Eyad

  • 3 years ago

Prove That :whatever the value of x: ___________________________________ \[sinx+cosx=\sqrt{2} [\cos(x-\Pi/4)]\]

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  1. experimentX
    • 3 years ago
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    \[ \sqrt{2}(\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x = \sqrt{2}(\cos \pi/4 \cos x + \sin \pi/4 \sin x ) \]

  2. experimentX
    • 3 years ago
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    \[ = \sqrt 2 \cos(\pi/4 -x) = \sqrt 2 \cos(x-\pi/4)\]

  3. eliassaab
    • 3 years ago
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    \[ \sin (x)+\cos (x)=\frac{e^{i x}-e^{-i x}}{2 i}+\frac{1}{2} \left(e^{i x}+e^{-i x}\right)=\\ \left(\frac{1}{2}+\frac{i}{2}\right) e^{-i x}+\left(\frac{1}{2}-\frac{i}{2}\right) e^{i x}=\\ \frac{e^{\frac{i \pi }{4}} e^{-i x}}{\sqrt{2}}+\frac{e^{\frac{1}{4} (-i) \pi } e^{i x}}{\sqrt{2}}=\\ \sqrt{2} \left(\frac{1}{2} e^{\frac{i \pi }{4}-i x}+\frac{1}{2} e^{i x-\frac{1}{4} (i \pi )}\right)=\sqrt 2 \cos\left( \frac \pi 4 -x\right) =\sqrt 2 \cos\left(x- \frac \pi 4 \right) \]

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