# PROVE THAT:- Cos^6 A + Sin^6 A= 1 - 3 × Sin^2 A × Cos^2 A

2
by VISHESH25
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You deserve it Shravani
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2014-07-09T16:29:24+05:30
As we know cos^2 +sin^2=1 now cubimg both side so {cos^2+ sin^2}^3= (1)^3
therefore by expanding we get cos^6 +sin^6+3cos^2 sin^4 +3sin^2 cos^4=1
therefore cos^6+sin^6 + 3cos^2sin^2 (cos^2+sin^2)=1
therefore cos^6+sin^6 +3cos^2 sin^2 =1
hence cos^6+sin^6= 1-3sin^2cos^2
pls mark as a best............. pls
Proof 1:sin^6 x + cos^6 x
= (sin^2 x + cos^2 x) ( sin^4 x - sin^2 cos^2 x + cos^4 x)
= (sin^4 x + cos^4 x) - sin^2 x cos^2 x
= (sin^2 x + cos^2 x)^2 - 2 sin^2 cos^2 x - sin^2 x cos^2 x
= 1 - 3 sin^2 x cos^2 x
Proof 2:
sin^6 x + cos^6 x
= (1 - cos^2 x)^3 + cos^6 x
= 1 - 3cos^2 x + 3cos^4 x - cos^6 x + cos^6 x
= 1 - 3cos^2 x(1-cos^2 x)
= 1 - 3 cos^2 x sin^2 x
Proof 1:
sin^6 x + cos^6 x
= (sin^2 x + cos^2 x) ( sin^4 x - sin^2 cos^2 x + cos^4 x)
= (sin^4 x + cos^4 x) - sin^2 x cos^2 x
= (sin^2 x + cos^2 x)^2 - 2 sin^2 cos^2 x - sin^2 x cos^2 x
= 1 - 3 sin^2 x cos^2 x
Proof 2:
sin^6 x + cos^6 x
= (1 - cos^2 x)^3 + cos^6 x
= 1 - 3cos^2 x + 3cos^4 x - cos^6 x + cos^6 x
= 1 - 3cos^2 x(1-cos^2 x)
= 1 - 3 cos^2 x sin^2 x
Proof 3:
sin^6 x + cos^6 x
= sin^6 x + cos^6 x + 3 sin^2 x cos^2 x - 3 sin^2 x cos^2 x
= sin^6 x + cos^6 s + 3 sin^2 x cos^2 x (sin^2 x + cos^2 x) - 3 sin^2 x cos^2 x
= 1 - 3sin^2 x cos^2 x
Proof 4:
sin^6 x + cos^6 x
= sin^6 x + cos^6 x - 1 + 1
= sin^6 x + cos^6 x - (sin^2 x + cos^2 x)^3 + 1
= sin^6 x + cos^6 x - sin^6 x - 3sin^4 x cos^2 x - 3 sin^2 x cos^4 x - sin^6 x + 1
= 1 - 3 sin^2 x cos^
2014-07-09T16:46:08+05:30

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[.·. (a³+b³)= (a+b) (a²-ab+b²) ]

[.·. Cos²A+Sin²A=1]

⇒1-3
[Hence, proved]

Proof 1:sin^6 x + cos^6 x
= (sin^2 x + cos^2 x) ( sin^4 x - sin^2 cos^2 x + cos^4 x)
= (sin^4 x + cos^4 x) - sin^2 x cos^2 x
= (sin^2 x + cos^2 x)^2 - 2 sin^2 cos^2 x - sin^2 x cos^2 x
= 1 - 3 sin^2 x cos^2 x
Proof 2:
sin^6 x + cos^6 x
= (1 - cos^2 x)^3 + cos^6 x
= 1 - 3cos^2 x + 3cos^4 x - cos^6 x + cos^6 x
= 1 - 3cos^2 x(1-cos^2 x)
= 1 - 3 cos^2 x sin^2 x
Proof 3:
sin^6 x + cos^6 x
= sin^6 x + cos^6 x + 3 sin^2 x cos^2 x - 3 sin^2 x cos^2 x
= sin^6 x + cos^6 s + 3 sin^2 x cos^2 x (sin^2 x + cos^2 x) - 3 sin^2 x cos^2 x
= 1 - 3sin^2 x cos^2 x
Proof 4:
sin^6 x + cos^6 x
= sin^6 x + cos^6 x - 1 + 1
= sin^6 x + cos^6 x - (sin^2 x + cos^2 x)^3 + 1
= sin^6 x + cos^6 x - sin^6 x - 3sin^4 x cos^2 x - 3 sin^2 x cos^4 x - sin^6 x + 1
= 1 - 3 sin^2 x cos^