Question

Solve tan-1 1/x+tan-1 1/2=π/4

Answer 1

Answer:

3

Step-by-step explanation:

Given :

tan⁻ ¹ 1/x + tan⁻ ¹ 1/2 = π/4

We know that

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

Replacing a = 1 / x and b = 1/2

⇒ tan⁻ ¹ 1/x + tan⁻ ¹ 1/2 = tan⁻ ¹ ( 1/x + 1/2 ) / ( 1 - 1/x × 1/2 )

⇒ tan⁻ ¹ 1/x + tan⁻ ¹ 1/2 = tan⁻ ¹ [ ( 2 + x ) / 2x ] / ( 1 - 1/2x )

⇒ tan⁻ ¹ 1/x + tan⁻ ¹ 1/2 = tan⁻ ¹ [ ( 2 + x ) / 2x ] / [ ( 2x - 1 ) / 2x ]

⇒ tan⁻ ¹ 1/x + tan⁻ ¹ 1/2 = tan⁻ ¹ ( 2 + x ) / ( 2x - 1 )

From given

⇒ π / 4 = tan⁻ ¹ ( 2 + x ) / ( 2x - 1 )

Taking tan on both sides

⇒ tan π/4 = tan[ tan⁻ ¹ ( 2 + x ) / ( 2x - 1 ) ]

⇒ 1 = ( 2 + x ) / ( 2x - 1 )

⇒ 2x - 1 = 2 + x

⇒ 2x - x = 2 + 1

⇒ x = 3

Therefore the value of x is 3.