Question

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

Answer 1

\[ \sqrt{2}(\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x = \sqrt{2}(\cos \pi/4 \cos x + \sin \pi/4 \sin x ) \]

Answer 2

\[ = \sqrt 2 \cos(\pi/4 -x) = \sqrt 2 \cos(x-\pi/4)\]

Answer 3

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