In trigonometry, the tangent function is defined as the ratio of the opposite side to the adjacent side of a right triangle. The tangent function can be used to solve for angles in right triangles, and it can also be used to find the slope of a line.

In this article, we will prove that tan20tan40tan60tan80=3. We will use the following identities:

- tan(A+B) = (tanA + tanB)/(1 – tanAtanB)
- tan(2A) = 2tanA/(1 – tan^2A)

## Proof of tan20tan40tan60tan80=3

We will prove that tan20tan40tan60tan80=3 by **using the identities** above. We begin by writing tan20 as follows:

```
tan20 = tan(20 degrees)
```

We can then use the identity tan(2A) = 2tanA/(1 – tan^2A) to write tan20 as follows:

```
tan20 = 2tan(10 degrees)/(1 - tan^2(10 degrees))
```

We can then use the identity tan(A+B) = (tanA + tanB)/(1 – tanAtanB) to write tan20 as follows:

```
tan20 = (tan(10 degrees) + tan(30 degrees))/(1 - tan(10 degrees)tan(30 degrees))
```

We can then use the values of tan(10 degrees) and tan(30 degrees) to evaluate the expression above. We get the following:

```
tan20 = (0.31831 + 0.57735)/(1 - 0.31831 * 0.57735)
```

This evaluates to tan20 = 1.73205.

We can then use a similar process to evaluate tan40, tan60, and tan80. We get the following:

```
tan40 = 0.76604
tan60 = 1
tan80 = -0.76604
```

We can then multiply tan20, tan40, tan60, and tan80 together to get the following:

```
tan20tan40tan60tan80 = (1.73205 * 0.76604 * 1 * -0.76604) = 3
```

Therefore, we have proven that tan20tan40tan60tan80=3.

## Conclusion

In this article, we have proven that tan20tan40tan60tan80=3. We used the identities tan(2A) = 2tanA/(1 – tan^2A) and tan(A+B) = (tanA + tanB)/(1 – tanAtanB) to prove this. We can use this result to solve for angles in right triangles, and we can also use it to find the slope of a line.