Given that \(\cos \frac{a}{2} = \frac{\sqrt{51}}{12}\), we need to find \(\tan \frac{a}{2}\).

We can use the identity that relates \(\tan \frac{a}{2}\) to \(\sin \frac{a}{2}\) and \(\cos \frac{a}{2}\):

\[
\tan \frac{a}{2} = \frac{\sin \frac{a}{2}}{\cos \frac{a}{2}}
\]

We already know that:

\[
\cos \frac{a}{2} = \frac{\sqrt{51}}{12}
\]

From the previous calculation, we found that:

\[
\sin \frac{a}{2} = \frac{\sqrt{93}}{12}
\]

Now, we can calculate \(\tan \frac{a}{2}\):

\[
\tan \frac{a}{2} = \frac{\frac{\sqrt{93}}{12}}{\frac{\sqrt{51}}{12}} = \frac{\sqrt{93}}{\sqrt{51}}
\]

This can be simplified further:

\[
\tan \frac{a}{2} = \sqrt{\frac{93}{51}} = \sqrt{\frac{31 \times 3}{17 \times 3}} = \sqrt{\frac{31}{17}}
\]

So, \(\tan \frac{a}{2} = \sqrt{\frac{31}{17}}\).

Would you like more details on any step, or do you have any other questions?

Here are 5 related questions:

1. How do you find \(\sin a\) given \(\cos a\)?
2. How do you derive the half-angle identities for \(\tan \frac{a}{2}\)?
3. What is the double-angle identity for \(\tan a\)?
4. How can you verify \(\tan \frac{a}{2} = \frac{\sin a}{1 + \cos a}\)?
5. What is the relationship between \(\tan \frac{a}{2}\) and the other trigonometric functions?

**Tip:** When dealing with trigonometric identities, always check if the expression can be simplified further to make calculations easier.

Learn how to calculate tan(a/2) when given cos(a/2) = sqrt(51)/12. This problem involves trigonometric identities and their applications. Suitable for high school students studying trigonometry.

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