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


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Misc 19 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‘₯ 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

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.