1. **Problem statement:** Find the cosine of angle $X$ in a right triangle $XVW$ where the right angle is at vertex $V$. The sides are $XV=36$, $XW=85$, and $VW=77$. 2. **Recall the definition of cosine in a right triangle:** $$\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$ For angle $X$, the adjacent side is $XV$ and the hypotenuse is $XW$. 3. **Apply the formula:** $$\cos(X) = \frac{XV}{XW} = \frac{36}{85}$$ 4. **Simplify the fraction if possible:** The greatest common divisor of 36 and 85 is 1, so the fraction is already in simplest form. 5. **Final answer:** $$\cos(X) = \frac{36}{85}$$ This is the cosine of angle $X$ expressed as a proper fraction.