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Misc 17 - Solve tan-1 (x/y) - tan-1 (x - y)/(x + y) - Miscellaneous

Misc. 17 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 2
Misc. 17 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 3

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Misc 17 Solve tan−1 (x/y) – tan−1 (x − y)/(x + y) is equal to (A) π/2 (B) π/3 (C) π/4 (D) (−3π)/4 We know that tan–1 x – tan–1 y = tan–1 ((𝐱 − 𝐲)/(𝟏 + 𝐱𝐲)) tan−1 x/y – tan−1 ((x − y)/(x + y)) = tan−1 [(x/y − (x − y )/(x + y))/(1 + (x/y) ((x − y )/(x + y)) )] Replacing x by 𝑥/𝑦 & y by (x − y)/(x + y) = tan−1 (((𝑥 ( 𝑥 + 𝑦 ) − 𝑦 ( 𝑥 − 𝑦 ))/(𝑦 ( 𝑥 + 𝑦 ) ))/((𝑦 ( 𝑥 + 𝑦 )+ 𝑥 ( 𝑥 − 𝑦))/(𝑦 ( 𝑥 + 𝑦) ))) = tan−1 ((𝑥 ( 𝑥 + 𝑦 ) − 𝑦 ( 𝑥 − 𝑦 ))/(𝑦 ( 𝑥 + 𝑦 ) ) × (𝑦 (𝑥 + 𝑦))/(𝑦 ( 𝑥 + 𝑦 )+ 𝑥 ( 𝑥 −𝑦) )) = tan−1 ((𝑥 ( 𝑥 + 𝑦 ) − 𝑦 ( 𝑥 − 𝑦 ))/(𝑦 ( 𝑥 + 𝑦 ) + 𝑥 (𝑥 + 𝑦)) × (𝑦 (𝑥 + 𝑦))/(𝑦 (𝑥 + 𝑦) )) = tan−1 ((𝑥 ( 𝑥 + 𝑦 ) − 𝑦 ( 𝑥 − 𝑦 ))/(𝑦 ( 𝑥 + 𝑦 ) + 𝑥 (𝑥 + 𝑦))) = tan−1 ((𝑥2+ 𝑥𝑦 − 𝑦𝑥 + 𝑦2)/(𝑦𝑥 +𝑦2 + 𝑥2 − 𝑥𝑦))) = tan−1 ((𝑥2+ 𝑦2+ 𝑦𝑥 − 𝑦𝑥)/(𝑦2 + 𝑥2 + 𝑦𝑥 − 𝑥𝑦))) = tan−1 ((𝑥2+ 𝑦2 )/(𝑦2 + 𝑥2 )) = tan−1 (1) = tan−1 ("tan " π/4) = π/4 Hence, Correct answer is C (As tan 𝜋/4 = 1)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.