Chứng minh rằng: ( sin³ x . sin³ x ) + ( cos³ x . cos³ x ) = 1 – 3sin² x . cos² x

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1. $VT=(\sin^3a.\sin^3a)+(\cos^3a.\cos^3a)$

   $=\cos^6a+\sin^6a$

   $=(\cos^2a)^3+(\sin^2a)^3$

   $=(\cos^2a+\sin^2a)(\cos^4a-\cos^2a.\sin^2a+\sin^4a)$

   $= \cos^4a+\sin^4a-\sin^2a\cos^2a$

   $= (\cos^2a+\sin^2a)^2-2\sin^2a\cos^2a-\sin^2a\cos^2a$

   $= 1-3\sin^2a\cos^2a$

   $= VP$

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2. Ta xét

   $VT = (\sin^3x . \sin^3x) + (\cos^3x . \cos^3x)$

   $= \sin^6x + \cos^6x$

   $= (\sin^2x)^3 + (\cos^2x)^3$

   $= (\sin^2x + \cos^2x)(\sin^4x + \cos^4x – \sin^2x \cos^2x)$

   $= \sin^4x + \cos^4x – \sin^2x \cos^2x$

   $= (\sin^2x)^2 + (\cos^2x)^2 + 2\sin^2x \cos^2x – 2\sin^2x \cos^2x – \sin^2x \cos^2x$

   $= (\sin^2x + \cos^2x)^2 – 3\sin^2x \cos^2x$

   $= 1 – 3\sin^2x \cos^2x = VP$.

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