Let's evaluate the expression [tex]\(1 - 2 \sin^2 (105^\circ)\)[/tex] step by step.
1. Calculate [tex]\(\sin(105^\circ)\)[/tex]:
The value of [tex]\(\sin(105^\circ)\)[/tex] is approximately [tex]\(0.9659258262890683\)[/tex].
2. Square [tex]\(\sin(105^\circ)\)[/tex]:
[tex]\[
\sin^2(105^\circ) \approx (0.9659258262890683)^2 = 0.9330127018922194
\][/tex]
3. Multiply by 2:
[tex]\[
2 \sin^2(105^\circ) \approx 2 \times 0.9330127018922194 = 1.8660254037844388
\][/tex]
4. Subtract from 1:
[tex]\[
1 - 2 \sin^2(105^\circ) \approx 1 - 1.8660254037844388 = -0.8660254037844388
\][/tex]
Thus, the value of [tex]\(1 - 2 \sin^2 (105^\circ)\)[/tex] is [tex]\(-0.8660254037844388\)[/tex].
This matches the known value of [tex]\(-\frac{\sqrt{3}}{2}\)[/tex], hence the correct answer is:
[tex]\[
-\frac{\sqrt{3}}{2}
\][/tex]
