Final answer:
After rewriting the equation tan(2x) = cot(x-18) by using the trigonometric identity tan(A+B), the solution for x is 36°, which corresponds to option B.
Step-by-step explanation:
To solve the equation tan(2x) = cot(x-18), we need to understand that the cotangent is the reciprocal of the tangent. This means that cot(x) = 1/tan(x). Using this, we can rewrite the equation as:
tan(2x) = 1/tan(x-18)
Which gives us:
tan(2x) * tan(x-18) = 1
Using the trigonometric identity tan(A+B) = (tan A + tan B)/(1 - tan A * tan B), we can equate 2x with A and (x-18) with B so that:
A + B = 2x + (x-18) = 3x - 18
For the identity to hold true, A + B must be an angle where the tangent is undefined, which occurs at 90° and its multiples. As such, we can write:
3x - 18 = 90
Solving for x:
3x = 108
x = 36°
Option B (x = 36°) is the correct answer as it satisfies both sides of the original equation.