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

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Misc 20 Using the fact that sin⁡(𝐴 + 𝐵)=sin⁡𝐴 cos⁡𝐵+cos⁡𝐴 sin⁡𝐵 and the differentiation, obtain the sum formula for cosines. Given sin⁡(𝐴 + 𝐵)=sin⁡𝐴 cos⁡𝐵+cos⁡𝐴 sin⁡𝐵 Consider A & B are function of 𝑥 Differentiating both side 𝑤.𝑟.𝑡.𝑥. 𝑑 sin﷮ 𝐴 + 𝐵﷯﷯﷯﷮𝑑𝑥﷯ = 𝑑 sin﷮𝐴﷯ cos﷮𝐵﷯ + cos⁡𝐴 sin⁡𝐵﷯﷮𝑑𝑥﷯ 𝑑 sin﷮ 𝐴 + 𝐵﷯﷯﷯﷮𝑑𝑥﷯ = 𝑑 sin﷮𝐴﷯ . cos⁡𝐵﷯﷮𝑑𝑥﷯ + 𝑑 cos﷮𝐴 ﷯. sin⁡𝐵﷯﷮𝑑𝑥﷯ cos 𝐴+𝐵﷯ . 𝑑 𝐴 + 𝐵﷯﷮𝑑𝑥﷯ = 𝑑 sin﷮𝐴﷯ . cos⁡𝐵﷯﷮𝑑𝑥﷯ + 𝑑 cos﷮𝐴 ﷯. sin⁡𝐵﷯﷮𝑑𝑥﷯ cos 𝐴+𝐵﷯ . 𝑑𝐴﷮𝑑𝑥﷯ + 𝑑𝐵﷮𝑑𝑥﷯﷯ = 𝑑 sin﷮𝐴﷯﷯﷮𝑑𝑥﷯. cos⁡𝐵 + 𝑑 cos﷮𝐵﷯﷯﷮𝑑𝑥﷯ 𝑠𝑖𝑛⁡A﷯ + 𝑑 cos﷮𝐴﷯﷯﷮𝑑𝑥﷯. 𝑠𝑖𝑛⁡𝐵 + 𝑑 sin﷮𝐵﷯﷯﷮𝑑𝑥﷯. 𝑐os A﷯ = cos﷮𝐴﷯. 𝑑𝐴﷮𝑑𝑥﷯. cos B − sin﷮𝐵﷯. 𝑑𝐵﷮𝑑𝑥﷯ sin﷮𝐴﷯ − sin﷮𝐴﷯. 𝑑𝐴﷮𝑑𝑥﷯. sin﷮𝐵﷯+ cos﷮𝐵﷯. 𝑑𝐵﷮𝑑𝑥﷯. 𝑐os A = cos﷮𝐴﷯. 𝑑𝐴﷮𝑑𝑥﷯. cos B − sin﷮𝐴﷯ . 𝑑𝐴﷮𝑑𝑥﷯ 𝑠𝑖𝑛⁡B − sin﷮𝐵﷯. 𝑑𝐵﷮𝑑𝑥﷯. 𝑠𝑖𝑛 𝐴﷮+ cos B﷯. 𝑑𝐵﷮𝑑𝑥﷯. 𝑐os A = 𝑑𝐴﷮𝑑𝑥﷯ cos﷮𝐴﷯ cos﷮𝐵﷯− sin﷮𝐴﷯ sin﷮𝐵﷯﷯ + 𝑑𝐵﷮𝑑𝑥﷯ − sin﷮𝐵﷯ sin﷮𝐴﷯+ cos﷮𝐵﷯ cos﷮𝐴﷯﷯ = cos﷮𝐴﷯ cos﷮𝐵﷯− sin﷮𝐴﷯ sin﷮𝐵﷯﷯ 𝑑𝐴﷮𝑑𝑥﷯ + 𝑑𝐵﷮𝑑𝑥﷯﷯ Thus, cos 𝐴+𝐵﷯ . 𝑑𝐴﷮𝑑𝑥﷯ + 𝑑𝐵﷮𝑑𝑥﷯﷯ = cos﷮𝐴﷯ cos﷮𝐵﷯− sin﷮𝐴﷯ sin﷮𝐵﷯﷯ 𝑑𝐴﷮𝑑𝑥﷯ + 𝑑𝐵﷮𝑑𝑥﷯﷯ cos 𝐴+𝐵﷯ . 𝑑𝐴﷮𝑑𝑥﷯ + 𝑑𝐵﷮𝑑𝑥﷯﷯﷮ 𝑑𝐴﷮𝑑𝑥﷯ + 𝑑𝐵﷮𝑑𝑥﷯﷯﷯ = cos﷮𝐴﷯ cos﷮𝐵﷯− sin﷮𝐴﷯ sin﷮𝐵﷯ cos 𝐴+𝐵﷯ = cos﷮𝐴﷯ cos﷮𝐵﷯+ sin﷮𝐴﷯ sin﷮𝐵﷯ Hence proved

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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