Ex 2.2, 15 - If tan-1 (x - 1)/(x - 2) + tan-1 (x+1)/(x+2) = pi/4

Ex 2.2, 15 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 2
Ex 2.2, 15 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 3

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Question 6 If tanβˆ’1 (x βˆ’ 1)/(x βˆ’ 2) + tanβˆ’1 (x + 1)/(x + 2) = πœ‹/4 , then find the value of x. Given tanβˆ’1 ((𝐱 βˆ’ 𝟏)/(𝐱 βˆ’ 𝟐)) + tanβˆ’1 ((𝐱 + 𝟏)/(𝐱 + 𝟐)) = πœ‹/4We know that tanβˆ’1 x + tanβˆ’1 y = tanβˆ’1 ((𝐱 + 𝐲 )/( 𝟏 βˆ’ 𝐱𝐲)) Replacing x by (π‘₯ βˆ’ 1)/(π‘₯ βˆ’ 2) and y by ((π‘₯ + 1)/(π‘₯ + 2)) tanβˆ’1 [((x βˆ’ 1 )/(x βˆ’ 2) + (x + 1)/(x + 2))/(1βˆ’ (x βˆ’ 1)/(x βˆ’ 2) Γ— (x + 1)/(x + 2))]=" " πœ‹/4 tanβˆ’1 [((x βˆ’ 1 )/(x βˆ’ 2) + (x + 1)/(x + 2))/(1βˆ’ (x βˆ’ 1)/(x βˆ’ 2) Γ— (x + 1)/(x + 2))]="tan " πœ‹/4 = tan-1 [(((x βˆ’ 1) (x + 2) + (x + 1)(x βˆ’ 2))/((x βˆ’ 2) (x + 2) ))/(((x βˆ’ 2) (x + 2) βˆ’ (x βˆ’ 1) (x + 1))/((x βˆ’ 2) (x + 2) ))]((x βˆ’ 1 )/(x βˆ’ 2) + (x + 1)/(x + 2))/(1βˆ’ (x βˆ’ 1)/(x βˆ’ 2) Γ— (x + 1)/(x + 2)) = "tan " 𝝅/πŸ’ (((x βˆ’ 1) (x + 2) + (x + 1)(x βˆ’ 2))/((x βˆ’ 2) (x + 2) ))/(((x βˆ’ 2) (x + 2) βˆ’ (x βˆ’ 1) (x + 1))/((x βˆ’ 2) (x + 2) )) = 1 ((x βˆ’ 1) (x + 2) + (x + 1)(x βˆ’ 2))/((x βˆ’ 2) (x + 2) ) Γ— ((x βˆ’ 2) (x + 2))/((x + 2) (x βˆ’ 2) βˆ’ (x βˆ’ 1)(x + 1)) = 1 ((x βˆ’ 1) (x + 2) + (x + 1)(x βˆ’ 2))/((x + 2) (x βˆ’ 2) βˆ’ (x βˆ’ 1)(x + 1)) = 1 Using (a + b) (a – b) = a2 – b2 ((x βˆ’ 1) (x + 2) + (x + 1)(x βˆ’ 2))/(π‘₯2 βˆ’ 22 βˆ’[π‘₯2 βˆ’ 12]) = 1 (π‘₯ (π‘₯ + 2) βˆ’ 1 (π‘₯ + 2) + π‘₯ (π‘₯ βˆ’ 2) + 1 (π‘₯ βˆ’ 2))/(π‘₯2 βˆ’ 4 βˆ’ π‘₯2 + 1) = 1 (π‘₯2 + 2π‘₯ βˆ’ π‘₯ βˆ’ 2 + π‘₯2 βˆ’ 2π‘₯ + π‘₯ βˆ’ 2 )/(π‘₯2 βˆ’ π‘₯2 βˆ’ 4 + 1) = 1 (2x2 βˆ’4)/(βˆ’3) = 1 2x2 – 4 = βˆ’3 2x2 = βˆ’3 + 4 2x2 = 1 x2 = 1/2 ∴ x = Β± 𝟏/√𝟐

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.