State whether the following are true or false. Justify your answer.
$(i)$ The value of $\tan A$ is always less than $1$.
$(ii)$ $\sec A = \frac{12}{5}$ for some value of angle $A$.

(A-D) $(i)$ False.
Consider a right-angled triangle where the side opposite to $\angle A$ is $12$ units and the side adjacent to $\angle A$ is $5$ units. Then $\tan A = \frac{12}{5} = 2.4$. Since $2.4 > 1$,the statement that $\tan A$ is always less than $1$ is false.
$(ii)$ True.
We know that $\sec A = \frac{\text{Hypotenuse}}{\text{Side adjacent to } \angle A}$.
Given $\sec A = \frac{12}{5}$,let the hypotenuse $AC = 12k$ and the adjacent side $AB = 5k$ for some positive constant $k$.
By Pythagoras theorem,$BC^2 = AC^2 - AB^2 = (12k)^2 - (5k)^2 = 144k^2 - 25k^2 = 119k^2$.
Thus,$BC = \sqrt{119}k \approx 10.9k$.
Since a triangle with sides $5k, 10.9k,$ and $12k$ satisfies the triangle inequality $(5k + 10.9k > 12k)$,such a triangle is possible. Therefore,the statement is true.