Hello, Kaitosan!
Start with the right side . . .$\displaystyle \sin x + \cos x \:=\:\sqrt{2}\,\sin(x + \tfrac{\pi}{4})$
$\displaystyle \sqrt{2}\,\sin(x + \tfrac{\pi}{4}) \;=\;\sqrt{2}\left(\sin x\cos\tfrac{\pi}{4} + \cos x\sin\tfrac{\pi}{4}\right)$
. . . . . . . . . $\displaystyle = \;\sqrt{2}\,\bigg[\sin x\cdot\frac{1}{\sqrt{2}} + \cos x\cdot\frac{1}{\sqrt{2}}\bigg] $
. . . . . . . . . $\displaystyle = \;\sin x + \cos x$
Actually... I'd rather you try with the left side (I have specific reasons). Can you do that....?
This is as far as I can go -
(sinx)^2 + (cosx)^2 = 1
(sinx + cosx)^2 = 1 - sin2x
sinx + cosx = sqrt(1 - sin2x)
.................................................. ............
Hi
This is not the good way to proceed.
$\displaystyle a \sin x + b \cos x = \sqrt{a^2+b^2} \left(\frac{a}{\sqrt{a^2+b^2}} \sin x + \frac{b}{\sqrt{a^2+b^2}} \cos x \right)$
$\displaystyle \frac{a}{\sqrt{a^2+b^2}}$ and $\displaystyle \frac{b}{\sqrt{a^2+b^2}}$ can be seen as the cos and the sin of an angle $\displaystyle \theta$ because the sum of their square is equal to 1
$\displaystyle a \sin x + b \cos x = \sqrt{a^2+b^2} \left(\sin x \cos \theta+ \cos x \sin \theta\right) = \sqrt{a^2+b^2} \:\sin (x + \theta)$
$\displaystyle \sin x+\cos x=\sin x+\sin\left(\frac{\pi}{2}-x\right)=$
$\displaystyle =2\sin\frac{x+\frac{\pi}{2}-x}{2}\cos\frac{x-\frac{\pi}{2}+x}{2}=$
$\displaystyle =2\sin\frac{\pi}{4}\cos\left(x-\frac{\pi}{4}\right)=$
$\displaystyle =2\frac{\sqrt{2}}{2}\cos\left(\frac{\pi}{4}-x\right)=\sqrt{2}\sin\left(\frac{\pi}{2}-\frac{\pi}{4}+x\right)=$
$\displaystyle =\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)$
Can't you just take Soroban's answer and play it in reverse.
For example. First take $\displaystyle \sqrt{2}$ out as a factor
$\displaystyle \sin x + \cos x = \sqrt{2}\left(\frac{1}{\sqrt{2}}\sin x + \frac{1}{\sqrt{2}}\cos x\right)$
Then observe that $\displaystyle \sin \frac{\pi}{4} = \cos \frac{\pi}{4}= \frac{1}{\sqrt{2}}$, and therefore
$\displaystyle \qquad = \sqrt{2}\left(\sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4}\right)$
And then from the compound angle formula for sin
$\displaystyle \qquad = \sqrt{2}\sin \left(x + \frac{\pi}{4}\right)$