Sure, let's find the exact value of [tex]\(\cos 75^\circ\)[/tex] expressed as [tex]\(\frac{\sqrt{2-\sqrt{3}}}{[?]}\)[/tex].
### Step-by-Step Solution:
1. Using Angle Addition Formula for Cosine:
[tex]\[
\cos(75^\circ) = \cos(45^\circ + 30^\circ)
\][/tex]
The angle addition formula for cosine is:
[tex]\[
\cos(A + B) = \cos A \cos B - \sin A \sin B
\][/tex]
2. Plug in the Specific Angle Values:
[tex]\[
\cos(75^\circ) = \cos(45^\circ) \cos(30^\circ) - \sin(45^\circ) \sin(30^\circ)
\][/tex]
3. Using Known Values for Trigonometric Functions at Standard Angles:
- [tex]\(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin(30^\circ) = \frac{1}{2}\)[/tex]
4. Substitute the Values:
[tex]\[
\cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)
\][/tex]
5. Simplify the Expression:
[tex]\[
\cos(75^\circ) = \frac{\sqrt{2} \cdot \sqrt{3}}{4} - \frac{\sqrt{2} \cdot 1}{4}
\][/tex]
[tex]\[
\cos(75^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}
\][/tex]
[tex]\[
\cos(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
### Compare with the Given Form:
The given form is:
[tex]\[
\cos 75^\circ = \frac{\sqrt{2 - \sqrt{3}}}{?}
\][/tex]
Currently, our derived expression is:
[tex]\[
\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
6. Equate to Convert to the Given Form:
We need to show that:
[tex]\[
\frac{\sqrt{2 - \sqrt{3}}}{?} = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
Given [tex]\(\cos 75^\circ = \frac{\sqrt{2 - \sqrt{3}}}{4}\)[/tex] in the alternative form, we confirm it by finding the value that fits in place of '[?]':
### Conclusion:
So, we conclude that:
[tex]\[
\boxed{4}
\][/tex]
is the value that fits in place of '[?]'. Thus, the exact value of [tex]\(\cos 75^\circ\)[/tex] expressed as [tex]\(\frac{\sqrt{2 - \sqrt{3}}}{4}\)[/tex] is correct.
