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


Transcript

Ex 2.2, 9 tan 1/2 [sinβˆ’1 2π‘₯/(1 + π‘₯2) + cosβˆ’1 (1 βˆ’ 𝑦2)/(1 + 𝑦2) ] We solve sinβˆ’1 2π‘₯/(1 + π‘₯2) & cosβˆ’1 ((1 βˆ’ 𝑦2)/(1 + 𝑦2)) separately Solving sinβˆ’1 πŸπ’™/(𝟏 + π’™πŸ) sinβˆ’1 2π‘₯/(1 + π‘₯2) Putting x = tan ΞΈ = sinβˆ’1 ((2 tanβ‘πœƒ)/(𝟏 + π’•π’‚π’πŸ 𝜽)) = sinβˆ’1 ((2 tanβ‘πœƒ)/(π’”π’†π’„πŸ 𝜽)) We write sin-1 & cosβˆ’1 in terms of tan-1 When there is 1 + x2 , we put x = tan ΞΈ = sinβˆ’1 ((2 sin⁑θ/cos⁑θ )/(1/cos2⁑θ )) = sinβˆ’1 (2 sin⁑θ/cos⁑θ Γ—cos2⁑θ/1) = sinβˆ’1 (𝟐 π’”π’Šπ’β‘πœ½ π’„π’π’”β‘πœ½ ) = sinβˆ’1 (sin 2ΞΈ) = 2ΞΈ ∴ sinβˆ’1 2π‘₯/(1 + π‘₯2) = 2ΞΈ Solving cosβˆ’1 ((𝟏 βˆ’ 𝐲𝟐)/(𝟏 + 𝐲𝟐)) cosβˆ’1 ((𝟏 βˆ’ 𝐲𝟐)/(𝟏 + 𝐲𝟐)) Putting y = tan 𝜢 = cosβˆ’1 ((1 βˆ’γ€– tan2〗⁑𝛼)/(𝟏 +γ€– π­πšπ§πŸγ€—β‘πœΆ )) = cosβˆ’1 ((1 βˆ’γ€– tan2〗⁑𝛼)/𝐬𝐞𝐜𝟐⁑𝜢 ) = cosβˆ’1 ((1 βˆ’ (𝑠𝑖𝑛2 𝛼 )/(π‘π‘œπ‘ 2 𝛼))/((1 )/(π‘π‘œπ‘ 2 𝛼))) = cosβˆ’1 (( (π‘π‘œπ‘ 2 𝛼 βˆ’ 𝑠𝑖𝑛2 𝛼 )/(π‘π‘œπ‘ 2 𝛼))/((1 )/(π‘π‘œπ‘ 2 𝛼))) = cosβˆ’1 ((π‘π‘œπ‘ 2 𝛼 βˆ’ 𝑠𝑖𝑛2 𝛼 )/(π‘π‘œπ‘ 2 𝛼)Γ—(π‘π‘œπ‘ 2 𝛼)/1) = cosβˆ’1 (π’„π’π’”πŸ 𝜢 βˆ’ π’”π’Šπ’πŸ 𝜢) = cosβˆ’1 (π’„π’π’”β‘πŸπœΆ ) = 2Ξ± Hence, cosβˆ’1 ((𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)) = 2Ξ± Now, we have sinβˆ’1 2π‘₯/(1 + π‘₯2) = 2ΞΈ & cosβˆ’1 ((1 βˆ’ 𝑦2)/(1 + 𝑦2)) = 2Ξ± Solving tan 𝟏/𝟐 ["sinβˆ’1 " πŸπ’™/(𝟏 + π’™πŸ) " + cosβˆ’1 " (𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)] Putting values = cosβˆ’1 (π’„π’π’”πŸ 𝜢 βˆ’ π’”π’Šπ’πŸ 𝜢) = cosβˆ’1 (π’„π’π’”β‘πŸπœΆ ) = 2Ξ± Hence, cosβˆ’1 ((𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)) = 2Ξ± Now, we have sinβˆ’1 2π‘₯/(1 + π‘₯2) = 2ΞΈ & cosβˆ’1 ((1 βˆ’ 𝑦2)/(1 + 𝑦2)) = 2Ξ± Solving tan 𝟏/𝟐 ["sinβˆ’1 " πŸπ’™/(𝟏 + π’™πŸ) " + cosβˆ’1 " (𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)] Putting values = tan 1/2 [2ΞΈ + 2Ξ±] = tan (ΞΈ + Ξ±) = (tan⁑θ + tan⁑𝛼)/(γ€–1 βˆ’ tan〗⁑θ tan⁑𝛼 ) = (𝒙 + π’š)/(𝟏 βˆ’ π’™π’š)

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.