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Misc 20 - Using the fact that sin (A + B) = sin A cos B - Miscellaneou

Misc 20 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Misc 20 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Misc 20 Using the fact that sin⁑(𝐴 + 𝐡)=sin⁑𝐴 cos⁑𝐡+cos⁑𝐴 sin⁑𝐡 and the differentiation, obtain the sum formula for cosines.Given sin⁑(𝐴 + 𝐡)=sin⁑𝐴 cos⁑𝐡+cos⁑𝐴 sin⁑𝐡 Consider A & B are function of π‘₯ Differentiating both side 𝑀.π‘Ÿ.𝑑.π‘₯. 𝑑(sin⁑(𝐴 + 𝐡) )/𝑑π‘₯ = 𝑑(sin⁑𝐴 cos⁑𝐡 + cos⁑𝐴 sin⁑𝐡)/𝑑π‘₯ 𝑑(sin⁑(𝐴 + 𝐡) )/𝑑π‘₯ = 𝑑(sin⁑𝐴 . cos⁑𝐡)/𝑑π‘₯ + 𝑑(cos⁑〖𝐴 γ€—. sin⁑𝐡)/𝑑π‘₯ cos (𝐴+𝐡) . 𝑑(𝐴 + 𝐡)/𝑑π‘₯ = 𝑑(sin⁑𝐴 . cos⁑𝐡)/𝑑π‘₯ + 𝑑(cos⁑〖𝐴 γ€—. sin⁑𝐡)/𝑑π‘₯ Using Product rule As (𝑒𝑣)’ = 𝑒’𝑣 + 𝑣’𝑒 𝒄𝒐𝒔 (𝑨+𝑩) . (𝒅𝑨/𝒅𝒙 + 𝒅𝑩/𝒅𝒙) = (𝑑(sin⁑𝐴 )/𝑑π‘₯. cos⁑𝐡" +" 𝑑(cos⁑𝐡 )/𝑑π‘₯ " " 𝑠𝑖𝑛⁑"A" ) + (𝑑(cos⁑𝐴 )/𝑑π‘₯. 𝑠𝑖𝑛⁑𝐡" +" 𝑑(sin⁑𝐡 )/𝑑π‘₯ ". " 𝑐"os A" ) = cos⁑𝐴.𝑑𝐴/𝑑π‘₯ ". cos B "βˆ’sin⁑𝐡.𝑑𝐡/𝑑π‘₯ " " sin⁑𝐴 βˆ’ sin⁑𝐴. 𝑑𝐴/𝑑π‘₯.sin⁑𝐡+cos⁑𝐡. 𝑑𝐡/𝑑π‘₯ ". " 𝑐"os A" = cos⁑𝐴.𝑑𝐴/𝑑π‘₯ ". cos B "βˆ’sin⁑𝐴 .𝑑𝐴/𝑑π‘₯ " " 𝑠𝑖𝑛⁑"B" βˆ’ sin⁑𝐡. 𝑑𝐡/𝑑π‘₯. 𝑠𝑖𝑛 𝐴⁑"+ cos B" . 𝑑𝐡/𝑑π‘₯ ". " 𝑐"os A" = 𝑑𝐴/𝑑π‘₯ (cos⁑𝐴 cosβ‘π΅βˆ’sin⁑𝐴 sin⁑𝐡 ) + 𝑑𝐡/𝑑π‘₯ (βˆ’sin⁑𝐡 sin⁑𝐴+cos⁑𝐡 cos⁑𝐴 ) (cos⁑𝐴 cosβ‘π΅βˆ’sin⁑𝐴 sin⁑𝐡 ) (𝑑𝐴/𝑑π‘₯ + 𝑑𝐡/𝑑π‘₯) Thus, cos (𝐴+𝐡) . (𝑑𝐴/𝑑π‘₯ + 𝑑𝐡/𝑑π‘₯) = (cos⁑𝐴 cosβ‘π΅βˆ’sin⁑𝐴 sin⁑𝐡 ) (𝑑𝐴/𝑑π‘₯ + 𝑑𝐡/𝑑π‘₯) 𝒄𝒐𝒔" " (𝑨+𝑩) = 𝒄𝒐𝒔⁑𝑨 π’„π’π’”β‘π‘©βˆ’π’”π’Šπ’β‘π‘¨ π’”π’Šπ’β‘π‘© Hence proved

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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.