Question

If tan theta + sin theta = a and tan theta - sin theta =b, then prove that a2 - b2 =4×rootab

Answer 1

tan x + sin x = a  

tan x - sin x = b  

ab = (tan x + sin x) (tan x - sin x)  

= tan² x - sin² x  

= (sin² x / cos² x) - sin² x  

= sin² x [1/cos² x - 1 ]  

= sin² x [sec² x - 1]  

= sin² x tan² x  

sin x tan x = √(ab) ... ... ... [1]  

tan x + sin x = a  

tan x - sin x = b  

tan x = (1/2) (a + b)  

sin x = (1/2) (a - b)  

tan x sin x = (1/4) (a² - b²) ... ... ... [2]  

From [1] and [2],  

a² - b² = 4√(ab)