Question

If A = 30°;
show that:
(sinA - cosA)² = 1 - sin2A

Answer 1

Given that A = 30°

LHS = ${\displaystyle {(\sin \text{A}–\cos \text{A})}^2}$

=${\displaystyle {(\sin 30°–\cos 30°)}^2}$

=${\displaystyle {(\frac{1}{2}–\frac{\sqrt{3}}{2})}^2}$

= ${\displaystyle \frac{1}{4}+\frac{3}{4}–\frac{\sqrt{3}}{2}}$

= ${\displaystyle 1 –\frac{\sqrt{3}}{2}}$

= ${\displaystyle 2–\frac{\sqrt{3}}{2}}$

RHS = 1 – sin 2A

= 1 – sin 2(30°)

= 1 – sin60°

= ${\displaystyle 1–\frac{\sqrt{3}}{2}}$

= ${\displaystyle \frac{2–\sqrt{3}}{2}}$

LHS = RHS