Is it true that $\Delta\cos(k\|x\|)=-k\left(k\cos(k\|x\|)+2\frac{\sin(k\|x\|)}{\|x\|}\right)$? As per the title: is it true that for $x\in\mathbb{R}^3$ and $k\in\mathbb{R}_{\ge 0}$,
$$\Delta\cos(k\|x\|)=-k\left(k\cos(k\|x\|)+2\frac{\sin(k\|x\|)}{\|x\|}\right)\qquad(\star)$$
My own calculations yield that
$$\begin{aligned}
\Delta\cos(k\|x\|)&=-k\nabla\left\{\sin(k\|x\|)\frac{x}{\|x\|}\right\}
\\
&=-k\left(k\cos(k\|x\|)\frac{x^2}{\|x\|^2}+\sin(k\|x\|)\frac{\|x\|+\frac{x^2}{\|x\|^3}}{\|x\|^2}\right)
\end{aligned}$$
which is not the same as $(\star)$. Note that $(\star)$ is the value that I am supposed to verify, as per a textbook I am following.
 A: The Laplacian is the sum of the unmixed second partial derivatives, 
$$\Delta f = \frac{\partial f}{\partial x_1^2} + \dots + \frac{\partial f}{\partial x_N^2}$$
and in this particular case,
$$f = \cos \left ( k \lVert x \rVert \right ) , \qquad k \in \mathbb{R} ,\; k \ge 0 ,\; x \in R^N $$
Let's examine the partial derivatives:
$$\frac{\partial f}{\partial x_1} = - \sin \left ( k \lVert x \rVert \right ) \frac{ k x_1 }{\lVert x \rVert}$$
All partial derivatives are similar. The second partial derivative of above is
$$\frac{\partial^2 f}{\partial x_1^2} = - \cos \left ( k \lVert x \rVert \right ) \frac{k^2 x_1^2}{\lVert x \rVert^2} - \sin \left ( k \lvert x \rVert \right ) \frac{k}{\lVert x \rVert} + \sin \left ( k \lVert x \rVert \right ) \frac{k x_1^2}{\lVert x \rVert^3}$$
Note the middle $\sin$ term: it does not have $x_1$ in it.
Summing all the unmixed second partial derivatives, to get the Laplacian, we get
$$\Delta \cos\left(k\lVert x \rVert\right) = -\cos\left(k\lVert x\rVert\right)\frac{k^2 \lVert x \rVert^2}{\lVert x \rVert^2} - \sin\left(k\lVert x\rVert\right)\frac{N k}{\lVert x \rVert} + \sin\left(k \lVert x \rVert\right)\frac{k \lVert x \rVert^2}{\lVert x \rVert^3}$$
Again, note the middle $\sin$ term. Because we have $N$ coordinates, and each unmixed second partial derivative had the same term in it, it occurs in the entire sum $N$ times.
Applying $$\frac{\lVert x \rVert^2}{\lVert x \rVert^2} = 1, \text{ and } \frac{\lVert x \rVert^2}{\lVert x \rVert^3} = \frac{1}{\lVert x \rVert}$$
we can simplify the sum (and Laplacian) to
$$\Delta \cos\left(k\lVert x \rVert\right) = -k^2 \cos\left(k\lVert x\rVert\right) - \sin\left(k\lVert x\rVert\right)\frac{k (N - 1)}{\lVert x \rVert}$$
If $x \in \mathbb{R}^3$, then obviously $N = 3$, and
$$\Delta \cos\left(k\lVert x \rVert\right) = -k \left ( k \cos\left(k\lVert x\rVert\right) + \frac{2 \sin\left(k\lVert x\rVert\right)}{\lVert x \rVert} \right )$$
