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if sin@+sin^2@=1 prove that cos^12@+3cos^10@+3cos^8@+cos^6@-1=0

if sin@+sin^2@=1 prove that cos^12@+3cos^10@+3cos^8@+cos^6@-1=0

Grade:12

2 Answers

Sunil Raikwar
askIITians Faculty 45 Points
7 years ago
given sin@=1-sin2@=cos2@
therefore sin6@+3sin5@+3sin4@+sin3@-1
={sin2@+sin@}2-1=1-1=0

Thanks & Regards
Sunil Raikwar
askIITian's Faculty

Rinkoo Gupta
askIITians Faculty 80 Points
7 years ago
sin@+sin^2@=1
=>sin@=1-sin^2@=cos^2@
Now
cos^12@+3cos^10@+3cos^8@+cos^6@-1
Using cos^2@=sin@ in this expression we get
sin^6@+cos^6@+3cos^10@+3cos^8@-1
=1-3sin^2@cos^2@+3cos^10@+3cos^8@-1
=-3sin^2@cos^2@+3cos^10@+3cos^8@
=-3sin^2@cos^2@+3sin^4@+3cos^10@
=3sin^2@(sin^2@-cos^2@)+3cos^10@
=-3sin^2@cos2@+3cos^10@
=-3sin^2@cos2@+3cos^4@cos^6@
=-3sin^2@cos2@+3sin^2@cos^6@
=3sin^2@(cos^6@-cos2@)
=3sin^2@(sin^2@cos^2@-cos2@)
=3sin^2@(cos^2@sin^2@-cos^2@+sin^2@)
=3sin^2@{cos^2@(sin^2@-1)+sin^2@}
=3sin^2@{-sin@cos^2@+sin^2@}
=3sin^2@{-sin@sin@+sin^2@}
=3sin^2@{=sin^2@+sin^2@}
=0
Thanks & Regards
Rinkoo Gupta
AskIITians Faculty

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