We need to determine the tangent line equation and the second derivative $\frac{d^2y}{dx^2}$ for the parametric curve given:

$x = \tan^2 t - 1, \quad y = \cot t, \quad t = -\frac{\pi}{4}.$

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Step 1: Find the point $(x, y)$ on the curve for $t = -\frac{\pi}{4}$.

Compute $x$:

$x = \tan^2\left(-\frac{\pi}{4}\right) - 1.$ $\tan\left(-\frac{\pi}{4}\right) = -1$, so: $x = (-1)^2 - 1 = 0.$

Compute $y$:

$y = \cot\left(-\frac{\pi}{4}\right).$ $\cot\left(-\frac{\pi}{4}\right) = -1$, so: $y = -1.$

Thus, the point is $(0, -1)$.

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Step 2: Find the slope of the tangent line ($\frac{dy}{dx}$).

From parametric equations, the slope is: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.$

Compute $\frac{dx}{dt}$:

$x = \tan^2 t - 1 \implies \frac{dx}{dt} = 2\tan t \sec^2 t.$

Compute $\frac{dy}{dt}$:

$y = \cot t \implies \frac{dy}{dt} = -\csc^2 t.$

Evaluate $\frac{dy}{dx}$ at $t = -\frac{\pi}{4}$:

At $t = -\frac{\pi}{4}$: $\tan\left(-\frac{\pi}{4}\right) = -1, \quad \sec^2\left(-\frac{\pi}{4}\right) = 2, \quad \csc^2\left(-\frac{\pi}{4}\right) = 2.$ $\frac{dx}{dt} = 2(-1)(2) = -4, \quad \frac{dy}{dt} = -2.$ $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-2}{-4} = \frac{1}{2}.$

The slope of the tangent line is $\frac{1}{2}$.

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Step 3: Write the equation of the tangent line.

The equation of the tangent line is: $y - y_1 = m(x - x_1),$ where $m = \frac{1}{2}$, and $(x_1, y_1) = (0, -1)$.

$y - (-1) = \frac{1}{2}(x - 0) \implies y + 1 = \frac{1}{2}x.$

Final tangent line equation: $y = \frac{1}{2}x - 1.$

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Part 2: Compute $\frac{d^2y}{dx^2}$.

From parametric equations: $\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \div \frac{dx}{dt}.$

Differentiate $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-\csc^2 t}{2\tan t \sec^2 t}.$ Simplify: $\frac{dy}{dx} = \frac{-\csc^2 t}{2\tan t (1 + \tan^2 t)}.$

Using the quotient rule, find $\frac{d}{dt}\left(\frac{dy}{dx}\right)$, then divide by $\frac{dx}{dt} = -4$. The full derivative simplifies at $t = -\frac{\pi}{4}$.

At $t = -\frac{\pi}{4}$, $\frac{d^2y}{dx^2}$ computes to $-\frac{3}{4}$.

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Final Results:

1. Tangent line equation: $y = \frac{1}{2}x - 1.$
2. $\frac{d^2y}{dx^2} = -\frac{3}{4}$.