Find the value of tan6° tan42° tan66° tan78°

To find the value of tan6° tan42° tan66° tan78°, we can use the trigonometric identity:

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)

Let's start by finding the value of tan6° tan42°:

tan(6° + 42°) = (tan6° + tan42°) / (1 - tan6° tan42°)

tan48° = (tan6° + tan42°) / (1 - tan6° tan42°)

Now, let's find the value of tan66° tan78°:

tan(66° + 78°) = (tan66° + tan78°) / (1 - tan66° tan78°)

tan144° = (tan66° + tan78°) / (1 - tan66° tan78°)

Since tan144° = tan(90° + 54°) = -tan54°, we can rewrite the equation as:

-tan54° = (tan66° + tan78°) / (1 - tan66° tan78°)

Now, we have two equations:

tan48° = (tan6° + tan42°) / (1 - tan6° tan42°)

-tan54° = (tan66° + tan78°) / (1 - tan66° tan78°)

We can solve these equations simultaneously to find the value of tan6° tan42° tan66° tan78°.