Yes, this statement is true. It uses Euler's formula, which states that for any real number $\theta$:

$e^{i\theta} = \cos(\theta) + i \sin(\theta).$

In the given equation:

$2 \cdot e^{\frac{-1}{12} \pi \cdot i} = 2 \cdot \left( \cos\left(-\frac{\pi}{12}\right) + i \sin\left(-\frac{\pi}{12}\right) \right).$

Here's how it works:

1. The expression $e^{\frac{-1}{12} \pi \cdot i}$ can be rewritten using Euler's formula: $e^{i\theta} = \cos(\theta) + i \sin(\theta).$ In this case, $\theta = -\frac{\pi}{12}$.

2. Substituting $\theta$: $e^{\frac{-1}{12} \pi \cdot i} = \cos\left(-\frac{\pi}{12}\right) + i \sin\left(-\frac{\pi}{12}\right).$

3. Multiplying both sides by 2 gives: $2 \cdot e^{\frac{-1}{12} \pi \cdot i} = 2 \cdot \left( \cos\left(-\frac{\pi}{12}\right) + i \sin\left(-\frac{\pi}{12}\right) \right).$

So, the equation holds true.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions for further exploration:

1. What is the general form of Euler's formula for complex numbers?
2. How does the sign of the angle affect the cosine and sine functions in Euler's formula?
3. Can Euler's formula be used to derive the trigonometric identities?
4. What are the applications of Euler's formula in electrical engineering?
5. How is the magnitude of a complex number derived from its exponential form?

Tip: Euler's formula is fundamental in converting between exponential and trigonometric forms of complex numbers, making calculations more manageable.