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

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experimentX
 2 years ago
Best ResponseYou've already chosen the best response.2\[ \sqrt{2}(\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x = \sqrt{2}(\cos \pi/4 \cos x + \sin \pi/4 \sin x ) \]

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.2\[ = \sqrt 2 \cos(\pi/4 x) = \sqrt 2 \cos(x\pi/4)\]

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1\[ \sin (x)+\cos (x)=\frac{e^{i x}e^{i x}}{2 i}+\frac{1}{2} \left(e^{i x}+e^{i x}\right)=\\ \left(\frac{1}{2}+\frac{i}{2}\right) e^{i x}+\left(\frac{1}{2}\frac{i}{2}\right) e^{i x}=\\ \frac{e^{\frac{i \pi }{4}} e^{i x}}{\sqrt{2}}+\frac{e^{\frac{1}{4} (i) \pi } e^{i x}}{\sqrt{2}}=\\ \sqrt{2} \left(\frac{1}{2} e^{\frac{i \pi }{4}i x}+\frac{1}{2} e^{i x\frac{1}{4} (i \pi )}\right)=\sqrt 2 \cos\left( \frac \pi 4 x\right) =\sqrt 2 \cos\left(x \frac \pi 4 \right) \]
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