Finding the Possible Values of tan(x/2) given tan(x) = 5/12 using the Double Angle Formula

To find the possible values of tan(x/2) when tan(x) = 5/12, we can use the double angle formula for tangent. The double angle formula for tangent is given by:

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

Let's break down the problem into smaller steps to solve it.

Step 1: Given Information
We are given that tan(x) = 5/12.

Step 2: Find the Value of tan(A)
To use the double angle formula, we need to find the value of tan(A). In this case, A is x/2. To find tan(A), we can use the half-angle formula for tangent, which is given by:

tan(A) = (1 - cos(A))/(sin(A))

Step 3: Finding sin(A) and cos(A)
To find sin(A) and cos(A), we can use the Pythagorean identity for sine and cosine:

sin^2(A) + cos^2(A) = 1

We know that:

tan(A) = (1 - cos(A))/(sin(A))

Rearranging the equation, we get:

1 - cos(A) = tan(A) * sin(A)

Squaring both sides of the equation, we get:

1 - 2cos(A) + cos^2(A) = tan^2(A) * sin^2(A)

Since sin^2(A) = 1 - cos^2(A), we can substitute this into the equation:

1 - 2cos(A) + cos^2(A) = tan^2(A) * (1 - cos^2(A))

Expanding and rearranging the equation, we get:

tan^2(A) * cos^2(A) + 2cos(A) - 1 = 0

This is a quadratic equation in cos(A). Solving this equation will give us the values of cos(A), and then we can find sin(A) using sin^2(A) = 1 - cos^2(A).

Step 4: Applying the Double Angle Formula
Now that we have the value of tan(A), we can apply the double angle formula for tangent:

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

Substituting the value of tan(A) into the formula, we get:

tan(2A) = (2 * (1 - cos(A))/(sin(A)))/(1 - (1 - cos(A))^2/(sin^2(A)))

Simplifying the expression, we get:

tan(2A) = (2 - 2cos(A))/(2sin(A)cos(A))

Step 5: Simplifying the Expression
We can simplify the expression further by canceling out common factors:

tan(2A) = 1/sin(A) - 1/cos(A)

Since we have the values of sin(A) and cos(A) from step 3, we can substitute these values into the equation to find the possible values of tan(2A).