Question:

1. (i) sin(A+B) = sin A + sin B
- The correct addition formula (proved in the textbook) is
$$\sin(A+B)=\sin A\cos B+\cos A\sin B.$$
- Since $\cos B$ and $\cos A$ are generally not equal to 1, the right‑hand side is not equal to $\sin A+\sin B$.
- Counter‑example: Take $A=30^{\circ}, B=60^{\circ}$. Then $\sin(90^{\circ})=1$, whereas $\sin30^{\circ}+\sin60^{\circ}=0.5+0.866=1.366
eq1$.
- Hence the statement is False.

2. (ii) The value of sin θ increases as θ increases
- From the unit‑circle definition, for $0^{\circ}\le θ\le 90^{\circ}$ the ordinate of the point on the circle increases with the angle; therefore $\sin θ$ is a strictly increasing function in this interval.
- In the textbook it is mentioned that "the sine of an acute angle increases as the angle increases".
- Hence, within the principal range $0^{\circ}\le θ\le 90^{\circ}$ the statement is True (it is false outside this range, but the question is understood in the context of acute angles).

3. (iii) The value of cos θ increases as θ increases
- In the same interval $0^{\circ}\le θ\le 90^{\circ}$ the abscissa of the point on the unit circle decreases as the angle increases; consequently $\cos θ$ is a decreasing function.
- Example: $\cos30^{\circ}=\frac{\sqrt3}{2}\approx0.866$, while $\cos60^{\circ}=\frac12=0.5$.
- Therefore the statement is False.

4. (iv) sin θ = cos θ for all values of θ
- Equality holds only when $\tan θ = 1$, i.e. $θ = 45^{\circ} + n\times180^{\circ}$ (or $θ = \frac{\pi}{4}+n\pi$ in radians).
- For most angles the two functions have different values (e.g., $\sin30^{\circ}=0.5$, $\cos30^{\circ}=0.866$).
- Hence the statement is False.

5. (v) cot A is not defined for A = 0°
- By definition $\cot A = \frac{\cos A}{\sin A}$.
- At $A=0^{\circ}$, $\sin0^{\circ}=0$, so the denominator becomes zero and the expression is undefined.
- Therefore the statement is True.

Summary of answers:
(i) False, (ii) True, (iii) False, (iv) False, (v) True.