Find tan(3 theta) in terms of tan theta

Use the formula
tan (a + b) = (tan a + tan b)/[1 - tan a tan b)
in two steps. First, let a = b = theta and get a formula for tan (2 theta).
tan (2 theta) = 2 tan theta/[(1 - tan theta)^2]
Then write down the equation for
tan (2 theta + theta)

Are you sure you are supposed to use complex numbers to answer this question? My previous answer used a trigonometric identity. I don't see a way to use complex numbers, but there probably is a way.

Exp(3 i theta) = [Exp(i theta)]^3 --->

cos(3 theta) = c^3 - 3cs^2

sin(3 theta) = 3c^2s - s^3

where c = cos(theta) and s = sin(theta)

This means that:

tan(3 theta) =

[ 3c^2s - s^3]/[c^3 - 3cs^2]

divide numerator and denominator by c^3:

tan(3 theta) =

[ 3t - t^3]/[1 - 3t^2] =

where t = tan(theta)

Thanks everyone for your help. Yes I did have to use complex numbers. Thanks Count I really should have seen your method for myself as I had found cos 3theta and sin 3 theta in an earlier problem.
Thanks again Jake

To find tan(3 theta) in terms of tan(theta), we can use the formula for the addition of angles in trigonometry. First, let's find tan(2 theta) using this formula:

tan(2 theta) = (tan(theta) + tan(theta)) / (1 - tan(theta) * tan(theta))

Simplifying this equation further, we have:

tan(2 theta) = 2 tan(theta) / (1 - tan(theta)^2)

Now, let's write down the equation for tan(2 theta + theta):

tan(2 theta + theta) = (tan(2 theta) + tan(theta)) / (1 - tan(2 theta) * tan(theta))

Substituting the value we obtained for tan(2 theta) in the previous step, we get:

tan(2 theta + theta) = (2 tan(theta) / (1 - tan(theta)^2) + tan(theta)) / (1 - (2 tan(theta) / (1 - tan(theta)^2)) * tan(theta))

Simplifying further, we have:

tan(2 theta + theta) = (3 tan(theta) - tan(theta)^3) / (1 - 3 tan(theta)^2)

Therefore, tan(3 theta) in terms of tan(theta) is equal to:

tan(3 theta) = (3 tan(theta) - tan(theta)^3) / (1 - 3 tan(theta)^2)