1. **State the problem:** We want to find the limit as $y$ approaches 1 of the function $$f(y) = \sec \left(y \sec^3 y - \tan^2 y - 1\right).$$ 2. **Evaluate the expression inside the secant:** We first calculate the limit of the inner expression inside the sec function as $y \to 1$: $$g(y) = y \sec^3 y - \tan^2 y - 1.$$ Substitute $y=1$: \begin{align*} \sec 1 &= \frac{1}{\cos 1}, \\ \tan 1 &= \tan 1 \text{ (use radian measure)}. \end{align*} So, $$g(1) = 1 \cdot \sec^3 1 - \tan^2 1 - 1 = \sec^3 1 - \tan^2 1 - 1.$$ 3. **Calculate the numerical values:** Approximate using radians: \begin{align*} \cos 1 &\approx 0.5403, \\ \sec 1 &\approx \frac{1}{0.5403} \approx 1.8508, \\ \tan 1 &\approx 1.5574. \end{align*} Calculate: \begin{align*} \sec^3 1 &\approx (1.8508)^3 = 6.336, \\ \tan^2 1 &\approx (1.5574)^2 = 2.4255, \end{align*} thus, $$g(1) \approx 6.336 - 2.4255 - 1 = 2.9105.$$ 4. **Evaluate the secant of this value:** $$\lim_{y \to 1} f(y) = \sec(g(1)) = \sec(2.9105).$$ Estimate: \begin{align*} \cos(2.9105) &\approx -0.9717, \\ \sec(2.9105) &= \frac{1}{\cos(2.9105)} \approx -1.0289. \end{align*} 5. **Conclusion:** The limit is approximately $$\boxed{-1.0289}.$$