Answer:
<em>The equation of the tangent line </em>
![y - \frac{\pi }{4} = (\frac{-\pi }{2} +1)( x - \frac{\pi }{4} )](https://tex.z-dn.net/?f=y%20-%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20%3D%20%28%5Cfrac%7B-%5Cpi%20%7D%7B2%7D%20%2B1%29%28%20x%20-%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20%29)
Step-by-step explanation:
<u><em>Step(i):-</em></u>
Given function y = x cot (x) ....(i)
Differentiating equation (i) with respective to 'x' , we get
![\frac{dy}{dx} = x (-Co sec^{2} (x)) +cot(x) (1)](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20x%20%28-Co%20sec%5E%7B2%7D%20%28x%29%29%20%2Bcot%28x%29%20%281%29)
<u><em>Step(ii):-</em></u>
The slope of the tangent line
![\frac{d y}{d x} = -x Co-sec^{2} (x) +cot x](https://tex.z-dn.net/?f=%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D%20-x%20Co-sec%5E%7B2%7D%20%28x%29%20%2Bcot%20x)
![(\frac{d y}{d x} )x_{=\frac{\pi }{4} } = -\frac{\pi }{4} Co-sec^{2} (\frac{\pi }{4} ) +cot \frac{\pi }{4}](https://tex.z-dn.net/?f=%28%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%29x_%7B%3D%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%7D%20%20%3D%20-%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20Co-sec%5E%7B2%7D%20%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%29%20%2Bcot%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D)
We will use trigonometry formulas
![Cosec(\frac{\pi }{4} ) = \sqrt{2}](https://tex.z-dn.net/?f=Cosec%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%29%20%3D%20%5Csqrt%7B2%7D)
![sec(\frac{\pi }{4} ) = \sqrt{2}](https://tex.z-dn.net/?f=sec%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%29%20%3D%20%5Csqrt%7B2%7D)
![Cot(\frac{\pi }{4} ) = 1](https://tex.z-dn.net/?f=Cot%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%29%20%3D%201)
Now the slope of the tangent
![\frac{dy}{dx} =-\frac{\pi }{4} (\sqrt{2})^{2} )+1](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D-%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%28%5Csqrt%7B2%7D%29%5E%7B2%7D%20%20%29%2B1)
![\frac{dy}{dx} =-\frac{\pi }{2} +1](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D-%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%2B1)
<u><em>Step(iii):-</em></u>
Given
Substitute
in y = x cot (x)
![y = \frac{\pi }{4} cot (\frac{\pi }{4} )](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20cot%20%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%29)
![y = \frac{\pi }{4}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D)
The point of the tangent line ![(x ,y ) = (\frac{\pi }{4} , \frac{\pi }{4} )](https://tex.z-dn.net/?f=%28x%20%2Cy%20%29%20%3D%20%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%2C%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%29)
<em>The equation of the tangent line </em>
![y - y_{1} = m ( x - x_{1} )](https://tex.z-dn.net/?f=y%20-%20y_%7B1%7D%20%3D%20m%20%28%20x%20-%20x_%7B1%7D%20%29)
![y - \frac{\pi }{4} = (\frac{-\pi }{2} +1)( x - \frac{\pi }{4} )](https://tex.z-dn.net/?f=y%20-%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20%3D%20%28%5Cfrac%7B-%5Cpi%20%7D%7B2%7D%20%2B1%29%28%20x%20-%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20%29)