How to find the constants of this transfer function? 
A stabil, casual and continuous system have the transfer function $$H(s)=\frac{a}{s+b}$$ If the input signal to the system, $x(t)=\sin(t)$, become the output signal $y(t)=\sqrt{2}\sin(t-\pi/4)$ when stabil. Find the constants $a$ and $b$.

How I solved this question was to start with taking the Laplace transform of $x(t)$ and $y(t)$ to get $X(s)$ and $Y(s)$. So I start with $x(t)$.
$$x(t)=\sin(t) \Rightarrow X(s)=\frac{1}{s^2+1}$$
Then $y(t)$
$$y(t)=\sqrt{2}\sin(t-\pi/4) = \sin(t)-\cos(t) \Rightarrow Y(s) = \frac{1-s}{s^2+1}$$
So now I have that $Y(s)=H(s)X(s)$ is equal to
$$\frac{1-s}{s^2+1} = \frac{a}{(s+b)(s^2+1)}$$
Now we can see that both $a=b=1-s$ because this is the only way for $Y(s)$ and $H(s)X(s)$ to be the same. But this is incorrect, according to the solution it should be that $a=2$ and $b=1$. How is this possible?

Well, we know that:
$$\mathcal{H}\left(\text{s}\right):=\frac{\text{Y}\left(\text{s}\right)}{\text{X}\left(\text{s}\right)}=\frac{\text{a}}{\text{s}+\text{b}}\tag1$$
Now, we also know that:
$$\frac{\text{Y}\left(\text{s}\right)}{\text{X}\left(\text{s}\right)}=\frac{\frac{1-\text{s}}{1+\text{s}^2}}{\frac{1}{1+\text{s}^2}}=1-\text{s}\tag2$$
So, we need to solve:
$$1-\text{s}=\frac{\text{a}}{\text{s}+\text{b}}\tag3$$
And we know that for a stable system the poles of $\mathcal{H}\left(\text{s}\right)$ has to lie left half of the complex plane.
