\text{If}\mathrm{ 5}\sin Y+\cos Y=\sqrt{2}\sin Y\text{, then find the value of }\tan Y.

To find the value of $\tan Y$, we start with the given trigonometric equation:

$5\sin Y+\cos Y=\sqrt{2}\sin Y$

We divide both sides of the equation by $\cos Y$ (assuming $\cos Y\ne 0$):

$\frac{5\sin Y}{\cos Y}+\frac{\cos Y}{\cos Y}=\sqrt{2}\frac{\sin Y}{\cos Y}$

Using the identity $\tan Y=\frac{\sin Y}{\cos Y}$, we can rewrite the equation as:

$5\tan Y+1=\sqrt{2}\tan Y$

Rearranging the equation to group the $\tan Y$ terms on one side:

$1=\sqrt{2}\tan Y-5\tan Y$

Factor out $\tan Y$ from the right-hand side:

$1=(\sqrt{2}-5)\tan Y$

Solve for $\tan Y$:

$\tan Y=\frac{1}{\sqrt{2}-5}$

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{2}+5$: