State whether the following are true of false. Justify your answer.
(i) \( sin (A + B) \ = \ sin A \ + \ sin B \).

(ii) The value of \( sin \theta \) increases as \( \theta \) increases.

(iii) The value of \( cos\theta \) increases as \( \theta \) increases.

(iv) \( sin\theta \ = \ cos\theta \) for all values of \( \theta \).

(v) cot A is not defined for A = 0°.

(i) False, because
for a case when A = 60° and B = 30°,

\( sin (A + B) \ \)
\( = \ sin (60° + 30°) \ \)
\( = \ sin 90° \ = \ 1 \)

and, \( sin A \ + \ sin B \ \)
\( = \ sin 60° \ + \ sin 30° \ \)
\( = \frac{ \sqrt{3}}{2} \ + \ \frac{1}{2} \ \)
\( = \ \frac{ \sqrt{3} + 1}{2} \)

\( sin (A + B) \ \ne \ sin A + sin B \)

(ii) True, as we can see from the table as the value of \( sin \theta \) increases \( \theta \) also increases.



(iii) False, as we can see from the table as the value of \( cos\theta \) increases \( \theta \) decreases.



(iv) False, as it is only true for \( \theta \ = \ 45° \).

(v) True, as \( tan 0° \ = \ 0 \) and \( cot0° \ = \ \frac{1}{tan0°} \ = \ \frac{1}{0} \) ,i.e., not defined.