1. **Problem statement:** Find the equation of the tangent line to the curve $y=\cos x$ at the point where $x=1$. 2. **Formula used:** The equation of the tangent line to a function $y=f(x)$ at $x=a$ is given by: $$y = f(a) + f'(a)(x - a)$$ where $f'(a)$ is the derivative of $f(x)$ evaluated at $x=a$. 3. **Find the derivative:** For $y=\cos x$, the derivative is: $$f'(x) = -\sin x$$ 4. **Evaluate the function and derivative at $x=1$:** $$f(1) = \cos 1$$ $$f'(1) = -\sin 1$$ 5. **Write the tangent line equation:** $$y = \cos 1 - \sin 1 (x - 1)$$ 6. **Explanation:** - We first find the slope of the tangent line by differentiating $\cos x$ to get $-\sin x$. - Then we evaluate the slope and the function value at $x=1$. - Finally, we use the point-slope form of a line to write the tangent line equation. **Final answer:** $$y = \cos 1 - \sin 1 (x - 1)$$