How do you simplify tan 4θ to trigonometric functions of a unit θ?

To simplify \( \tan 4\theta \) using trigonometric functions of a unit angle \( \theta \), we can use the double angle formula for tangent. The formula states:

Double Angle Formula

The double angle formula for tangent is:

\( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \)

Applying the Formula

First, we express \( \tan 4\theta \) as:

\( \tan 4\theta = \tan(2 \cdot 2\theta) \)

Now, we can apply the double angle formula:

\( \tan 4\theta = \frac{2\tan 2\theta}{1 - \tan^2 2\theta} \)

Finding \( \tan 2\theta \)

Next, we need to find \( \tan 2\theta \) using the same formula:

\( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \)

Substituting Back

Now, substitute \( \tan 2\theta \) back into the equation for \( \tan 4\theta \):

• Replace \( \tan 2\theta \) in the first equation:
• \( \tan 4\theta = \frac{2 \left( \frac{2\tan \theta}{1 - \tan^2 \theta} \right)}{1 - \left( \frac{2\tan \theta}{1 - \tan^2 \theta} \right)^2} \)

Simplifying Further

This expression can be simplified further, but it involves algebraic manipulation. The final result will express \( \tan 4\theta \) in terms of \( \tan \theta \).

In summary, by using the double angle formulas, you can express \( \tan 4\theta \) in terms of \( \tan \theta \), allowing for simplification in various trigonometric problems.