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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

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

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

= (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

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^

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^

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[.·. (a³+b³)= (a+b) (a²-ab+b²) ]

⇒

⇒

⇒

[.·. Cos²A+Sin²A=1]

⇒1-3

[Hence, proved]

= (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

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^