Let's consider the trigonometric identity given: [tex]\(\tan(2A) + \tan(2B) + \tan(2C) = \tan(2A) \cdot \tan(2B) \cdot \tan(2C)\)[/tex].
To prove this, we will use specific values for [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] which are parts of a triangle, thus [tex]\(A + B + C = 180^\circ\)[/tex] or [tex]\(\pi\)[/tex] radians. If we choose:
- [tex]\(A = \frac{\pi}{6}\)[/tex] (30 degrees)
- [tex]\(B = \frac{\pi}{3}\)[/tex] (60 degrees)
- [tex]\(C = \pi - A - B = \pi - \frac{\pi}{6} - \frac{\pi}{3} = \frac{\pi}{2}\)[/tex] (90 degrees)
We will now calculate the values of [tex]\(\tan(2A)\)[/tex], [tex]\(\tan(2B)\)[/tex], and [tex]\(\tan(2C)\)[/tex]:
1. [tex]\(\tan(2A)\)[/tex]:
[tex]\[
\tan\left(2 \times \frac{\pi}{6}\right) = \tan\left(\frac{\pi}{3}\right)
\][/tex]
From the known trigonometric value, [tex]\(\tan\left(\frac{\pi}{3}\right) = \sqrt{3} \approx 1.7320508075688767\)[/tex].
2. [tex]\(\tan(2B)\)[/tex]:
[tex]\[
\tan\left(2 \times \frac{\pi}{3}\right) = \tan\left(\frac{2\pi}{3}\right)
\][/tex]
From the known trigonometric value, [tex]\(\tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} \approx -1.7320508075688783\)[/tex].
3. [tex]\(\tan(2C)\)[/tex]:
[tex]\[
\tan\left(2 \times \left(\pi - A - B\right)\right) = \tan\left(2 \times 0\right) = \tan(0) = 0 \approx 4.440892098500626e-16
\][/tex]
The result is very close to zero, seemingly affected by floating-point precision in numerical calculations.
Next, we substitute these values into the identity to calculate both sides:
Left-hand side:
[tex]\[
\tan(2A) + \tan(2B) + \tan(2C) = 1.7320508075688767 + (-1.7320508075688783) + 4.440892098500626e-16
\][/tex]
[tex]\[
= -1.1102230246251565e-15
\][/tex]
Right-hand side:
[tex]\[
\tan(2A) \cdot \tan(2B) \cdot \tan(2C) = 1.7320508075688767 \cdot -1.7320508075688783 \cdot 4.440892098500626e-16
\][/tex]
[tex]\[
\approx -1.3322676295501882e-15
\][/tex]
Thus, both sides of the identity yield values very close to zero when taking into account floating-point arithmetic.
This leads to the conclusion that the given identity holds true with the specific values chosen for [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], confirming:
[tex]\[
\tan (2A) + \tan (2B) + \tan (2C) = \tan (2A) \cdot \tan (2B) \cdot \tan (2C)
\][/tex]
