Does $\cos(x+y)=\cos x + \cos y$?

Find the value using a calculator: $\cos 75°$

At first I thought all I need is to separate the simpler known values like this:

$\cos 75^\circ = \cos 30°+\cos45° = {\sqrt3}/{2} + {\sqrt2}/{2} $

$= {(\sqrt3+\sqrt2)}/{2} $ This is my answer which translates to= $1.5731$ by calculator

However, when I used the calculator directly on $\cos 75°$, I get $0.2588$.

Where I am going wrong?

You used a formula $\cos(x+y) = \cos(x) + \cos(y)$ which is false. The correct formula is: $$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$$

The answer is simply: No and to see this take $y=0$ we find $$\cos(x)=\cos(x)+1$$ which's obviously false.

You can simply plot $cos(x+y)-(cos(x)+cos(y))$ to have your answer:

Alternatively: cos(A+B) = cosAcosB-sinAsinB with A = 45 and B is 30

As mentioned in other answers, $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$.

We also have $\cos(x)+\cos(y)=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$.

Since $\cos(0)=1$, we get $\cos(0+0)=1\ne2=\cos(0)+\cos(0)$.