Calculating Power of Sinusoidal Term and a Convolution expression
The power spectrum $P(e^{j\omega})$ is the Fourier transform of the autocorrelation function of $x(n)$:

$$R_x(k)=E\{x(n)x(n+k)\}\tag{1}$$

If you evaluate $(1)$ you should get

$$R_x(k)=\sigma_v^2\delta(k)+\frac{A^2}{2}\cos(k\omega_0)\tag{2}$$

Taking the Fourier transform of $(2)$ leads to the given expression for $P(e^{j\omega})$.

The convolution is obtained by noting that

$$u_0(\omega-\omega_0)\star W_B(e^{j\omega})=W_B(e^{j(\omega-\omega_0)})\tag{3}$$

and

$$\frac{1}{2\pi}\sigma_v^2\star W_B(e^{j\omega})=\sigma_v^2\frac{1}{2\pi}\int_{-\pi}^{\pi}W_B(e^{j\omega})d\omega=\sigma_v^2\tag{4}$$