Trigonometric Identity

We know that cos2x=cos2xsin2x\cos { 2x } =\cos ^{ 2 }{ x } -\sin ^{ 2 }{ x }

This is just a proof I have made. I don't know if it already exists but I just wanted to share it with you.

(eix)2=e2ix(eix)2=ei×2x(cosx+isinx)2=cos2x+isin2xcos2x+i2sin2x+2icosxsinx=cos2x+isin2xcos2x+(1)sin2x+i×2cosxsinx=cos2x+isin2xcos2xsin2x+isin2x=cos2x+isin2xSo,cos2x=cos2xsin2x{ \left( { e }^{ ix } \right) }^{ 2 }={ e }^{ 2ix }\\ \\ { \left( { e }^{ ix } \right) }^{ 2 }={ e }^{ i\times 2x }\\ \\ { \left( \cos { x } +i\sin { x } \right) }^{ 2 }=cos2x+isin2x\\ \\ \cos ^{ 2 }{ x } +{ i }^{ 2 }\sin ^{ 2 }{ x } +2i\cos { x } \sin { x } =cos2x+isin2x\\ \\ \cos ^{ 2 }{ x } +\left( -1 \right) \sin ^{ 2 }{ x } +i\times 2\cos { x } \sin { x } =cos2x+isin2x\\ \\ \cos ^{ 2 }{ x } -\sin ^{ 2 }{ x } +i\sin { 2x } =cos2x+isin2x\\ \\ So,\quad \cos { 2x } =\cos ^{ 2 }{ x } -\sin ^{ 2 }{ x }

Note by Archit Boobna
4 years, 8 months ago

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1 vote

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This also exists !!

Parth Bhardwaj - 4 years, 8 months ago

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But I thought it myself.

Archit Boobna - 4 years, 8 months ago

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Ok ! Did you try all of my problems - new ones ?

Parth Bhardwaj - 4 years, 8 months ago

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Nice observational skill Archit :)

A Former Brilliant Member - 4 years, 8 months ago

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Thanks!

Archit Boobna - 4 years, 8 months ago

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