This does not make sense, shouldn't the vector projection for $\vec v'$ be $\cos(\theta)v' + \sin(\theta)v'$ it seems to me that $\vec n\times \vec v$ and $\vec v$ are being projected unto $\vec v'$, not the other way around. My question is: why is the equation for $\vec v'$ in the image below, why is it correct? If you are trying to find the projection of $\vec v'$ on the $\vec n\times\vec v$ and the $\vec v$ axes shouldn't the equation be $\vec v'=cos(\theta)v' + sin(\theta)v'$
Note that you are missing some direction vectors in your equation. $$\vec{v}'=v'\cos\theta \hat i+v'\sin\theta\hat j$$ Here $\hat i$ is the unit vector along $\vec v$, and $\hat j$ is the perpendicular. You can write these as$$\hat i=\frac{\vec v}{|\vec v|}\\\hat j=\frac{\vec n \times \vec v}{|\vec n \times \vec v|}$$
Now all you need to know that $\vec n$ is perpendicular to $\vec v$, so that $|\vec n \times \vec v|=|\vec v|$. Also, the rotation does not change the length of the vector, so $|\vec v|=|\vec v'|=v'$.
If you put everything together, you get the equation in the figure.