Step 1: Use the given $\tan x$ to find $\sin x$ and $\cos x$. Given $\tan x = \frac{12}{5}$. In a right-angled triangle, $\tan x = \frac{\text{Opposite}}{\text{Adjacent}}$. Let the opposite side be $12k$ and the adjacent side be $5k$ for some constant $k$. Using the Pythagorean theorem, the hypotenuse $H$ is: $$H = \sqrt{(12k)^2 + (5k)^2} = \sqrt{144k^2 + 25k^2} = \sqrt{169k^2} = 13k$$ Now, we can find $\sin x$ and $\cos x$: $$\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{12k}{13k} = \frac{12}{13}$$ $$\cos x = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{5k}{13k} = \frac{5}{13}$$ Step 2: Substitute the values of $\sin x$ and $\cos x$ into the expression. The expression is $\frac{\sin x + 2 \cos x}{1 - \sin x}$. Substitute $\sin x = \frac{12}{13}$ and $\cos x = \frac{5}{13}$: $$\frac{\frac{12}{13} + 2 \left(\frac{5}{13}\right)}{1 - \frac{12}{13}}$$ Step 3: Simplify the numerator. $$\frac{12}{13} + 2 \left(\frac{5}{13}\right) = \frac{12}{13} + \frac{10}{13} = \frac{12+10}{13} = \frac{22}{13}$$ Step 4: Simplify the denominator. $$1 - \frac{12}{13} = \frac{13}{13} - \frac{12}{13} = \frac{13-12}{13} = \frac{1}{13}$$ Step 5: Divide the simplified numerator by the simplified denominator. $$\frac{\frac{22}{13}}{\frac{1}{13}} = \frac{22}{13} \times \frac{13}{1} = 22$$ The value of the expression is $\boxed{22}$. Send me the next one 📸