Is there a trigonometric function explaining $\cos(x)+\sin(x)$? if not, what is $\cos(x)+\sin(x)$ as a function of $\cos(x)$?
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$\begingroup$$$\sqrt 2 \cos (x-\pi/4)$$
By the Prostaphaeresis-Simpson-formula. If you need, you can convert sinuses into cosinuses via $\sin x =\cos(\pi/2-x)$.
Actually there is another fun way to see that this formula is true.
As you may know, if you take a point on the plane, located at an angle x with respect to the reference axis, then $(\cos x, \sin x)$ are proportional to the Cartesian coordinates of the point.
So your formula is is proportional to the sum of the coordinates of the point. This is basically equivalent (up to a proportionality constant) to compute a coordinate of your point with respect to an orthogonal Cartesian system that is rotated by an angle of 45 degrees.
$\endgroup$ 2 $\begingroup$We know that $\cos{x}+\sin{x}=\cos{x}+\cos{(\frac\pi2-x)}$.
You may keep it as it is, or you may apply the formula for $\cos{C}+\cos{D}$ according to your needs.
Another representation may also be $\cos{x}+\sin{x}=\cos{x}+\sqrt{1-\cos^2{x}}$. But in this case, as N.F. Taussig pointed out, we will have cases (with respect to in which quadrant $x$ lies) according to when to take the negative root, and when to take the positive one.
Does that help?
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