\text{If}14\sin Y+\cos Y=\sqrt{14}\sin Y\text{, then find the value of}\tan Y.

The correct option is:
$$\frac{-14-\sqrt{14}}{182}$$

Step-by-Step Derivation:

We are given the trigonometric equation:
$$14\sin Y+\cos Y=\sqrt{14}\sin Y$$
To find the value of $\tan Y$, we can start by gathering all the terms containing $\sin Y$ on one side of the equation.

Subtract $14\sin Y$ from both sides:
$$\cos Y=\sqrt{14}\sin Y-14\sin Y$$

Factor out $\sin Y$ on the right-hand side:
$$\cos Y=(\sqrt{14}-14)\sin Y$$

Since $\tan Y=\frac{\sin Y}{\cos Y}$, we divide both sides by $\cos Y$ (assuming $\cos Y\ne 0$):
$$1=(\sqrt{14}-14)\tan Y$$

Now, solve for $\tan Y$:
$$\tan Y=\frac{1}{\sqrt{14}-14}$$

To simplify this expression and match the given options, we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $\sqrt{14}+14$:
$$\tan Y=\frac{1·(\sqrt{14}+14)}{(\sqrt{14}-14)(\sqrt{14}+14)}$$

Using the difference of squares formula, $(a-b)(a+b)=a^2-b^2$, in the denominator:
$$\tan Y=\frac{14+\sqrt{14}}{{\left(\sqrt{14}\right)}^2-{14}^2}$$

Simplify the terms:
$$\tan Y=\frac{14+\sqrt{14}}{14-196}$$

Calculate the denominator:
$$14-196=-182$$

Substitute this back into the equation:
$$\tan Y=\frac{14+\sqrt{14}}{-182}$$

To make the denominator positive, we multiply both the numerator and the denominator by -1:
$$\tan Y=\frac{-14-\sqrt{14}}{182}$$

This matches the correct option exactly.