The curve $C$ is given by the equations

The line $y=x$ intersects with $C$ at the point $P$, as shown on the diagram.

Express the equation of the tangent to $C$ at the point $P$ in the form

where the greatest common divisor of $a,b,c$ is $1$.

Don't find $t$ - instead, notice that, at $P$, we have

This allows you to find the exact values of $\sin (t)$ and $\cos (t)$ at $P$ (you could, for example, square both sides, or draw a triangle).

When finding the gradient at $P$, notice that

at $P$, which allows you to find the exact value of $\cot (t)$.

Give the value of $a+b+c$.