Find the equation of the circle touching the line $(x-2)\cos\theta+(y-2)\sin\theta=1$ for all values of $\theta$ 
Find the equation of the circle touching the line $(x-2)\cos\theta+(y-2)\sin\theta=1$ for all values of $\theta$

The answer is given on toppr website.
It says, $(x-2)\cos\theta+(y-2)\sin\theta=\cos^2\theta+\sin^2\theta$, then comparing coefficients, it says,
$x-2=\cos\theta, y-2=\sin\theta$
I am not sure about this step. Do you think this step is valid?
Another answer exists on sarthaks website.
It says equation of tangent to the circle $(x-h)^2+(y-k)^2=a^2$ at $(x_1,y_2)$ is
$(x-h)x_1+(y-k)y_1=a^2$
Is this valid? I am not sure about this.

You can see that the family of lines is always at a distance $1$ from $(2,2)$.
Hence this family is tangent to $(x-2)^2+(y-2)^2=1$
