Question

The value of (sin 30° + cos 30°)² – (sin 60° – cos 60°)² is __________.

Answer 1

$$\begin{gathered} {\left(\sin 30°+\cos 30°\right)}^2-{\left(\sin 60°-\cos 60°\right)}^2 \\ ={\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\right)}^2-{\left(\frac{\sqrt{3}}{2}-\frac{1}{2}\right)}^2 \\ =\frac{{\left(\sqrt{3}+1\right)}^2-{\left(\sqrt{3}-1\right)}^2}{4} \end{gathered}$$
$$\begin{gathered} =\frac{\left(3+1+2\sqrt{3}\right)-\left(3+1-2\sqrt{3}\right)}{4} \\ =\frac{4+2\sqrt{3}-4+2\sqrt{3}}{4} \\ =\frac{4\sqrt{3}}{4} \\ =\sqrt{3} \end{gathered}$$

The value of (sin 30° + cos 30°)² – (sin 60° – cos 60°)² is $\underline{\sqrt{3}}$.