Need help with trigonometric question

Question

I've been trying to do the following question and spent an hour on it, so far I have no luck doing it. (the 29 is in degrees)

(Tan29 + TanX)/(1-Tan29TanX) = Tan2X

 

The question asks me to find the X value, I have no idea how to determine it, some help would help!

Answer 1

Recall the angle sum formula for tangents

 

tan(a+b) = (tan a + tan b)/(1 - tan a tan b)

 

Now let's use it here.

 

(tan 29 + tan x)/(1 - tan 29 tan x) = tan 2x

 

tan(29 + x) = tan 2x

 

29 + x + 180k = 2x

 

x = 29 + 180k ; k∈Z

 

 

 

 

Answer 2

The value of x must be 29 degrees. Here is why.

 

Tan (2X) = 2 Tan (x)/(1 - tan^2 X)     (one of the identities)

 

If x = 29 we have  Tan (58) = 2 Tan (29)/(1 - tan^2 29)

Tan (29) = Tan (X) so Tan (29) Tan (X) = Tan^2 (29)

Also Tan (29) + Tan (X) = 2 Tan (29).