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$$R_x(\theta)= \begin{bmatrix} \cos \left( \frac{\theta}{2} \right) & -i\sin \left( \frac{\theta}{2} \right) \\ -i\sin \left( \frac{\theta}{2} \right) & \cos \left( \frac{\theta}{2} \right)\end{bmatrix}$$

$$R_y(\theta)= \begin{bmatrix} \cos \left( \frac{\theta}{2} \right) & \sin \left( \frac{\theta}{2} \right) \\ \sin \left( \frac{\theta}{2} \right) & \cos \left( \frac{\theta}{2} \right)\end{bmatrix}$$

$$R_z(\theta)=\begin{bmatrix} \exp\left(-i\frac{\theta}{2}\right) & 0 \\ 0 & \exp\left(-i\frac{\theta}{2}\right)\end{bmatrix}$$

This is the rotation gates as matrices on the Bloch sphere. It was easy to show that $R_z$ to be the rotation about $z$ axis on the Bloch sphere. However, I can't find a way to show that $R_x$ and $R_y$ are rotations about $x$ and $y$ axis respectively and I can't find any solutions on the google.... Could anyone show me why $R_x$ and $R_y$ are rotations about $x$ and $y$ axis.

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I will leave $R_y(\theta)$ as an exercise for you.

Below is a verification of $R_x(\theta)$ is a rotation about the $x$-axis.

Denote the Pauli matrices as $$X=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, Y=\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, Z=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $$

We have the following expression for $R_x(\theta)$ $$R_x(\theta) = \cos \left( \frac{\theta}{2} \right)I-i\sin\left( \frac{\theta}{2}\right)X$$

Consider $$\rho=\frac12(I+r_xX+r_yY+r_zZ)$$

$\begin{align}\rho' &= R_x(\theta)\rho R_x(\theta)^\dagger \\ &=\frac12(I+r_xR_x(\theta)XR_x(\theta)^\dagger+r_yR_x(\theta)YR_x(\theta)^\dagger+r_zR_x(\theta)ZR_x(\theta)^\dagger)\end{align}\tag{1}$

Now, let's analyze each separate term:

\begin{align}R_x(\theta)XR_x(\theta)^\dagger &= \left(\cos \left( \frac{\theta}{2} \right)I-i\sin\left( \frac{\theta}{2}\right)X\right)X\left(\cos \left( \frac{\theta}{2} \right)I+i\sin\left( \frac{\theta}{2}\right)X\right)\\ &=\left(\cos \left( \frac{\theta}{2} \right)X-i\sin\left( \frac{\theta}{2}\right)I\right)\left(\cos \left( \frac{\theta}{2} \right)I+i\sin\left( \frac{\theta}{2}\right)X\right)\\ &=\cos^2 \left( \frac{\theta}{2}\right)X-i\sin \left(\frac{\theta}{2} \right)\cos \left(\frac{\theta}{2} \right)I+i\sin \left(\frac{\theta}{2} \right)\cos \left(\frac{\theta}{2} \right)I+\sin^2 \left( \frac{\theta}{2}\right)X\\ &= X\end{align}

\begin{align}R_x(\theta)YR_x(\theta)^\dagger &= \left(\cos \left( \frac{\theta}{2} \right)I-i\sin\left( \frac{\theta}{2}\right)X\right)Y\left(\cos \left( \frac{\theta}{2} \right)I+i\sin\left( \frac{\theta}{2}\right)X\right)\\ &=\left(\cos \left( \frac{\theta}{2} \right)Y+\sin\left( \frac{\theta}{2}\right)Z\right)\left(\cos \left( \frac{\theta}{2} \right)I+i\sin\left( \frac{\theta}{2}\right)X\right)\\ &=\cos^2 \left( \frac{\theta}{2}\right)Y+\sin \left(\frac{\theta}{2} \right)\cos \left(\frac{\theta}{2} \right)Z+\sin \left(\frac{\theta}{2} \right)\cos \left(\frac{\theta}{2} \right)Z-\sin^2 \left( \frac{\theta}{2}\right)Y\\ &= \cos (\theta) Y + \sin(\theta)Z\end{align}

\begin{align}R_x(\theta)ZR_x(\theta)^\dagger &= \left(\cos \left( \frac{\theta}{2} \right)I-i\sin\left( \frac{\theta}{2}\right)X\right)Z\left(\cos \left( \frac{\theta}{2} \right)I+i\sin\left( \frac{\theta}{2}\right)X\right)\\ &=\left(\cos \left( \frac{\theta}{2} \right)Z-\sin\left( \frac{\theta}{2}\right)Y\right)\left(\cos \left( \frac{\theta}{2} \right)I+i\sin\left( \frac{\theta}{2}\right)X\right)\\ &=\cos^2 \left( \frac{\theta}{2}\right)Z-\sin \left(\frac{\theta}{2} \right)\cos \left(\frac{\theta}{2} \right)Y-\sin \left(\frac{\theta}{2} \right)\cos \left(\frac{\theta}{2} \right)Y-\sin^2 \left( \frac{\theta}{2}\right)Z\\ &= -\sin (\theta) Y + \cos(\theta)Z\end{align}

\begin{align}\rho'&=\frac12 \left(I+r_xX+r_y(\cos (\theta) Y + \sin(\theta)Z)+r_z(-\sin (\theta) Y + \cos(\theta)Z)\right)\\ &= \frac12 \left(I+r_xX+(r_y(\cos (\theta) - r_z \sin (\theta)) Y + (r_y\sin(\theta) + r_z\cos(\theta))Z\right)\\\end{align} Hence if we write

$$\rho'=\frac12 \left(I+r_x'X+r_y'Y+r_z'Z\right)$$

$$\begin{bmatrix}r_x' \\ r_y' \\ r_z'\end{bmatrix}=\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \theta & - \sin \theta\\ 0 & \sin \theta & \cos \theta\end{bmatrix}\begin{bmatrix}r_x \\ r_y \\ r_z\end{bmatrix}$$

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