To determine which expression is equivalent to [tex]\(\sin(20^\circ)\cos(80^\circ) - \cos(20^\circ)\sin(80^\circ)\)[/tex], we can use the sine difference formula. The sine difference identity states:
[tex]\[
\sin(A)\cos(B) - \cos(A)\sin(B) = \sin(A - B)
\][/tex]
In the given problem, we have:
- [tex]\(A = 20^\circ\)[/tex]
- [tex]\(B = 80^\circ\)[/tex]
Using the sine difference identity, we can rewrite the expression:
[tex]\[
\sin(20^\circ)\cos(80^\circ) - \cos(20^\circ)\sin(80^\circ) = \sin(20^\circ - 80^\circ)
\][/tex]
Next, we calculate the angle inside the sine function:
[tex]\[
20^\circ - 80^\circ = -60^\circ
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\sin(20^\circ)\cos(80^\circ) - \cos(20^\circ)\sin(80^\circ) = \sin(-60^\circ)
\][/tex]
Therefore, the expression is equivalent to [tex]\(\sin(-60^\circ)\)[/tex].
The correct answer is:
[tex]\[
\sin \left(-60^{\circ}\right)
\][/tex]
