Question

Question 14: Solve tan¯¹ 1-x/1+x = 1/2 tan¯¹ x, (x > 0)

Class 12 - Math - Inverse Trigonometric Functions

Answer 1

tan inverse 1-x/1+x=1/2tan inverse x
2tan inverse 1-x/1+x=tan inverse x
tan inverse [2(1-x/1+x)]/1-(1-x/1+x)^2]=tan inverse x
multiplying tan on both sides we have [2(1-x/1+x)]/1-(1-x/1+x)^2]=x
On solving on pen paper I got x=+ or -1/1.732

Answer 2

 tan¯¹ 1-x/1+x = 1/2 tan¯¹ x, (x > 0)
 

2tan¯¹  (1+x / 1-x) =  tan¯¹x

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

⇒ 2(1-x / 1+x) / {(1+x)² - (1-x)²} / (1+x)²
⇒ x = 2-2x / [(1+x² +2x -1-x² + 2x) / (1+x) ]
⇒ x = 2-2x / [ 4x / 1+x ]
⇒ x = 2-2x (1+x) / 4x
⇒ 2 + 2x -2x - 2x² = 4x²
⇒ 2x² + 4x² -2 = 0
⇒ 6x² - 2 = 0
⇒ 2 (3x²-1) =0

⇒3x² -1 =0
⇒ 3x² = 1
⇒ x² = 1/3
⇒ x = +- 1/√3 

given that x>0 
∴ x = +1/√3

Hope helps!!!