To find the exact value of [tex]\(\tan \left(67.5^{\circ}\right)\)[/tex], let's break it down step by step using trigonometric identities and known values.
First, recognize that [tex]\(67.5^{\circ} = 45^{\circ} + 22.5^{\circ}\)[/tex]. We can use the tangent addition formula:
[tex]\[
\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A) \tan(B)}
\][/tex]
Here, [tex]\(A = 45^{\circ}\)[/tex] and [tex]\(B = 22.5^{\circ}\)[/tex].
1. Calculate [tex]\(\tan(45^{\circ})\)[/tex]:
[tex]\[
\tan(45^{\circ}) = 1
\][/tex]
2. Calculate [tex]\(\tan(22.5^{\circ})\)[/tex]:
We use the relationship:
[tex]\[
\tan(22.5^{\circ}) = \sqrt{\frac{1 - \sin(45^{\circ})}{1 + \sin(45^{\circ})}}
\][/tex]
Knowing that [tex]\(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
\tan(22.5^{\circ}) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}}
\][/tex]
Simplify the fraction:
[tex]\[
\tan(22.5^{\circ}) = \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}}
\][/tex]
3. Use the tangent addition formula:
Now we apply the formula:
[tex]\[
\tan(67.5^{\circ}) = \frac{\tan(45^{\circ}) + \tan(22.5^{\circ})}{1 - \tan(45^{\circ}) \tan(22.5^{\circ})}
\][/tex]
Substituting the values:
[tex]\[
\tan(67.5^{\circ}) = \frac{1 + \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}}}{1 - 1 \cdot \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}}}
\][/tex]
Given the simplifications and the known results:
[tex]\[
\tan(67.5^{\circ}) = 2.414213562373096
\][/tex]
Given the options, the correct answer can be identified as:
[tex]\[
\tan(22.5^{\circ}) = \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}}
\][/tex]
Therefore, the exact value we are looking for:
[tex]\[
\sqrt{\frac{2 + \sqrt{2}}{2 - \sqrt{2}}}
\][/tex]
[tex]\[
\boxed{\sqrt{\frac{2+\sqrt{2}}{2-\sqrt{2}}}}
\][/tex]
