[Trigonometric Identities]

Simplify the expression:

$$
\cos a \cdot \cos 2 a \cdot \cos 4 a \cdot \ldots \cdot \cos 2^{\mathrm{n}-1} a
$$
Multiply and divide the given expression by $\sin a$ and use the double angle formula.

## Otvet

For $a \neq \cdot n\left(\mathrm{n}\right.$ - integer) the expression is equal to $\sin \left(2^{\mathrm{n}} a\right) /\left(2^{\mathrm{n}} \sin a\right)$, for a=2 $\cdot n$ the expression is equal to 1, for a= $+2 \cdot n$ the expression is equal to -1 (assuming $n>1$ ).