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 = $\dfrac{12}{5}$ for some value of angle A.

(iii) cos A is the abbreviation used for the cosecant of angle A.

(iv) cot A is the product of cot and A.

(v) sin θ = $\dfrac{4}{3}$ for some angle θ.

(i) We know that,

tan 45° = 1.

∴ tan A is not always less than 1.

Hence, above statement is false.

(ii) We know that,

sec A ≥ 1

Since, $\dfrac{12}{5}$ is greater than 1.

∴ sec A = $\dfrac{12}{5}$ is possible.

Hence, above statement is true.

(iii) We know that,

cosine A is also referred as cos A.

Hence, above statement is false.

(iv) We know that,

cot A ≠ cot × A

Hence, above statement is false.

(v) We know that,

0 ≤ sin θ ≤ 1.

Since, $\dfrac{4}{3} \gt 1$

∴ sin θ cannot be equal to $\dfrac{4}{3}$.

Hence, above statement is false.