A curve's coordinates are given by equations : $y=e^t\sin t$, $x=e^t\cos t.$The relation between the parameter ' $t$ ' and the angle $\alpha$ between the tangent to the given curve and the $X$-axis is given by which of the following?

Given: Curve coordinates: $y=e^t\sin t, x=e^t\cos t.$

$$\begin{gathered} y=e^t\sin t \\ \Rightarrow \frac{dy}{dt}=e^t[\cos t+\sin t] \end{gathered}$$

$$\begin{gathered} x=e^t\cos t \\ \Rightarrow \frac{dx}{dt}=e^t[\cos t-\sin t] \end{gathered}$$

Therefore,

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{\cos t+\sin t}{\cos t-\sin t} \\ \Rightarrow \frac{\cos t+\sin t}{\cos t-\sin t}=\tan \alpha \\ \Rightarrow \frac{1+\tan t}{1-\tan t}=\tan \alpha \end{gathered}$$

$\Rightarrow \tan \left(\frac{\pi }{4}+t\right)=\tan \alpha$

$$\begin{gathered} \Rightarrow \left(\frac{\pi }{4}+t\right)=\alpha \\ \Rightarrow t=\alpha -\frac{\pi }{4} \end{gathered}$$

Solution is $\left(3\right).$