# Halving the Tangent, Twice

**Geometry**Level 5

If

\[ \large 5(\cos A+\cos C)+4(\cos A\cos C+1)=0, \]

then find the value of \( \tan\left(\frac A2\right) \tan\left(\frac C2\right)\).

If

\[ \large 5(\cos A+\cos C)+4(\cos A\cos C+1)=0, \]

then find the value of \( \tan\left(\frac A2\right) \tan\left(\frac C2\right)\).

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