To find the exact value of [tex]\(\sin 165^{\circ}\)[/tex], we can use trigonometric identities and angle decomposition.
1. First, note that [tex]\(165^{\circ}\)[/tex] can be written as [tex]\(180^{\circ} - 15^{\circ}\)[/tex].
2. Using the identity [tex]\(\sin(180^{\circ} - x) = \sin x\)[/tex], we get:
[tex]\[
\sin 165^{\circ} = \sin 15^{\circ}
\][/tex]
3. Next, we need to find the exact value of [tex]\(\sin 15^{\circ}\)[/tex]. We use the angle difference identity for sine:
[tex]\[
\sin(45^{\circ} - 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} - \cos 45^{\circ} \sin 30^{\circ}
\][/tex]
4. Substituting the known values:
[tex]\[
\sin 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \cos 30^{\circ} = \frac{\sqrt{3}}{2}, \quad \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \sin 30^{\circ} = \frac{1}{2}
\][/tex]
5. Now, calculate [tex]\(\sin 15^{\circ}\)[/tex]:
[tex]\[
\sin 15^{\circ} = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right)
\][/tex]
[tex]\[
\sin 15^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}
\][/tex]
[tex]\[
\sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
6. Therefore, the exact value of [tex]\(\sin 165^{\circ}\)[/tex] is:
[tex]\[
\sin 165^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
Hence, the correct choice is:
[tex]\[
\boxed{d. \frac{\sqrt{6} - \sqrt{2}}{4}}
\][/tex]
