To express the given trigonometric expression as a sine or cosine of an angle, we can use one of the angle subtraction identities. The hint provided suggests using the sine angle subtraction formula:
[tex]\[
\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B
\][/tex]
Given the expression:
[tex]\[
\sin 42^{\circ} \cos 17^{\circ} - \cos 42^{\circ} \sin 17^{\circ}
\][/tex]
We can identify it as fitting the form of the sine angle subtraction identity:
[tex]\[
\sin(A - B) = \sin A \cos B - \cos A \sin B
\][/tex]
By comparing terms, we see that:
- [tex]\( A = 42^{\circ} \)[/tex]
- [tex]\( B = 17^{\circ} \)[/tex]
Therefore, we substitute [tex]\( A \)[/tex] and [tex]\( B \)[/tex] into the identity:
[tex]\[
\sin 42^{\circ} \cos 17^{\circ} - \cos 42^{\circ} \sin 17^{\circ} = \sin(42^{\circ} - 17^{\circ})
\][/tex]
Perform the subtraction inside the sine function:
[tex]\[
42^{\circ} - 17^{\circ} = 25^{\circ}
\][/tex]
Thus, the given expression can be written as:
[tex]\[
\sin 42^{\circ} \cos 17^{\circ} - \cos 42^{\circ} \sin 17^{\circ} = \sin 25^{\circ}
\][/tex]
So the expression is:
[tex]\[
\sin 25^{\circ}
\][/tex]
