4.43 Given the fraction

$\frac{5}{1+\sqrt[3]{32 \cos ^{4} 15^{\circ}-10-8 \sqrt{3}}}$

Simplify the expression under the cube root, and then reduce the fraction.

## Group 6

Prove the identities (4.44-4.57):


4.43 Using formula (4.17), we find

$$
\begin{aligned}
& 32 \cos ^{4} 15^{\circ}-10-8 \sqrt{3}=32\left(\frac{1+\cos 30^{\circ}}{2}\right)^{2}-10-8 \sqrt{3}= \\
& =8\left(1+\frac{\sqrt{3}}{2}\right)^{2}-10-8 \sqrt{3}=4
\end{aligned}
$$

Next, we obtain

$$
\frac{5}{1+\sqrt[3]{4}}=\frac{1+4}{1+\sqrt[3]{4}}=\frac{(1+\sqrt[3]{4})(1-\sqrt[3]{4}+\sqrt[3]{16})}{1+\sqrt[3]{4}}=1-\sqrt[3]{4}+\sqrt[3]{16}
$$

We used the formula $a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)$, where $a$ and $b$ were 1 and $\sqrt[3]{4}$, respectively.