How to prove a trigonometric equation with the compound angle formula?

Question

Given that cos(x-30)=2sin(x+30), show that tanx=(2-3^1/2)/(1-2•3^1/2)

When I work through the first equation and manipulate it to get tanx, I get tanx=(3^1/2-2)/(2•3^1/2). I’m just not sure where I’ve gone wrong. Any help would be greatly appreciated :) Also all brackets are for ease of telling what is the numerator and denominator but not actually stated in the question.

Answer 1

I would guess that you've dropped a sin(x) in your expansion somewhere.

Using the sum and difference of angles formulas for sin and cos yield the equation cos(x)cos(30) + sin(x)sin(30) = 2(sin(x)cos(30) + cos(x)sin(30)). After evaluating with the unit circle, we get √3⁄2cos(x) + 1⁄2sin(x) = 2(√3⁄2sin(x) + 1⁄2cos(x)). Distributing the right side gives √3sin(x) + cos(x). We can multiply both sides to get rid of fractions, so √3cos(x) + sin(x) = 2√3sin(x) + 2cos(x). It's the sin(x) on the left that leads to the missing part of your answer. Rearranging by trig function gives (2-√3)cos(x) = (1-2√3)sin(x), and finally dividing both sides by cos(x) and flipping the sides so tan(x) is on the left yields tan(x) = (2-√3)/(1-2√3). You'll also find the values in the numerator to be switched from your answer, but that can be fixed by multiplying both the numerator and denominator by -1 (since multiplying by -1/-1 is the same as multiplying by 1)