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

  1. Class 12
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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)) )] = 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 + 0)/(𝑦2 + 𝑥2 + 0)) = tan-1 ((𝑥2+ 𝑦2 )/(𝑦2 + 𝑥2 )) = tan-1 (1) = tan-1 (tan π/4) = π/4 Hence, correct answer is C

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