The equation of a tangent to the hyperbola, 4 x ^ 2 - 5 y ^ 2 = 20,...

The slope of a line $ax + by + c = 0$ is $m = - \dfrac{a}{b} .$

Thus, the slope of the line $x - y = 2$ is $m = - \dfrac{1}{\left(- 1\right)} = 1$ and we know that if two lines are parallel then their slopes are equal, hence the slope of the tangent is also $m = 1 .$

The given hyperbola is $\dfrac{{x}^{2}}{5} - \dfrac{{y}^{2}}{4} = 1$

The equation of the tangent to the hyperbola $\dfrac{{x}^{2}}{{a}^{2}} - \dfrac{{y}^{2}}{{b}^{2}} = 1$ having slope $m$ is $y = mx \pm \sqrt{{a}^{2}{m}^{2} - {b}^{2}} .$

Hence, the equation of its tangent having slope $1$ is $y = x \pm \sqrt{5 \times {1}^{2} - 4}$

$⇒ y = x \pm 1$

Thus, the tangents are $x - y + 1 = 0 \text{ and } x - y - 1 = 0 .$