1. Lulama was asked to evaluate the following expression without the use of a calculator: \( \sin \left(180^{\circ}-135^{\circ}\right) \). She answered as follows: \[ \begin{array}{l} \operatorname{Sin}\left(180^{\circ}-135^{\circ}\right)=\sin 180^{\circ}-\sin 135^{\circ} \\ =0-\sin \left(180^{\circ}-45^{\circ}\right) \text { (step 1) } \\ \text { (step2) } \\ = \text { (step 3) } \\ =-\frac{\sin 45^{\circ}}{2} \text { (step 4) }\end{array} \] \( \begin{array}{ll}\text { 1.1. What law or property did Lulama use to write step 1? } & \text { (1) }\end{array} \) 1.2 Do you think Lulama is correct? Motivate your answer?

1.1. Lulama used a faulty “distributive” property of sine. In other words, she assumed that   sin(180° – 135°) = sin(180°) – sin(135°). There is no law that permits distributing the sine function over subtraction. 1.2. Lulama is not correct. The correct formula for the sine of a difference is   sin(α – β) = sin(α)·cos(β) – cos(α)·sin(β). Applying this to 180° – 135° would yield   sin(180° – 135°) = sin(180°)·cos(135°) – cos(180°)·sin(135°). Since sin(180°) = 0 and cos(180°) = –1, this simplifies to   0·cos(135°) – (–1)·sin(135°) = sin(135°). In fact, we know that sin(135°) = sin(180° – 45°) = sin(45°) = (√2)/2. Thus, the method Lulama used (writing sin(180° - 135°) as sin(180°) - sin(135°)) is not valid.