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

  1. Chapter 2 Class 12 Inverse Trigonometric Functions (Term 1)
  2. Serial order wise

Transcript

Ex 2.2, 15 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

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.