if tan a = 1/5 and tan b= 1/239, find the value tan (4a-b)?

Question

* USE DOUBLE ANGLE FORMULA
* answer in the book = 1

Answer 1

Looks like we'll have to use the double angle formula
several times.
tan(4a-b) = (tan 4a - tan b)/(1+tan 4a tan b).
So we need tan 4a.
tan 2a = 2 tan a/(1- tan² a) = 2/5 / (1- 1/25) = 2/5 * 25/24 = 5/12
tan 4a = 10/12 / (1- 25/144) = 10/12 * 144/119 = 120/119
So tan(4a-b) = (120/119 - 1/239)/ (1+ [120/119 * 1/239]) =
28561/28561 = 1.