The equation of the curve is given by:
[tex]y=5+\cot(x)-2\csc(x)[/tex]
Differentiating both sides of the equation with respect to x, we have:
[tex]\frac{dy}{dx}=2\cot(x)\csc(x)-\csc^2(x)[/tex]
Therefore, the slope of the tangent is given by the value of dy/dx when x= π / 2
[tex]2\cot(\frac{\pi}{2})\csc(\frac{\pi}{2})-\csc^2(\frac{\pi}{2})=-1[/tex]
Using the point slope formula, it follows that:
[tex]\begin{gathered} y-3=-1(x-\frac{\pi}{2}) \\ y=-x+\frac{\pi}{2}+3 \end{gathered}[/tex]
Therefore, the equation of the tangent at P is given by:
y = -x + π /2 + 3
