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Alisiya [41]
3 years ago
11

Prove that: (2-sin(2x))(sin(x) + cos(x)) = 2(sin^3(x) + cos^3(x))

Mathematics
1 answer:
Ber [7]3 years ago
5 0
   
\text{We use formulas: }\\  \\ 
1) ~~  (a + b)(a^2 -ab + b^2)   =a^3  + b^3 \\  \\ 
2)~~ \sin(2x) = 2\sin x \cos x  \\ \\ 
 3)~~ 1 =\sin^2(x) + cos^2(x) \\  \\ 
\text{We solve:} \\  \\ 
\Big(2-\sin(2x)\Big)\Big(\sin(x) + \cos(x)\Big) = 2\Big(\sin^3(x) + cos^3(x)\Big) \\  \\ 
\Big(2-2\sin(x)\cos(x)\Big)\Big(\sin(x) + \cos(x)\Big) = 2\Big(\sin^3(x) + cos^3(x)\Big) \\  \\ 
2\Big(1-\sin(x)\cos(x)\Big)\Big(\sin(x) + \cos(x)\Big) = 2\Big(\sin^3(x) + cos^3(x)\Big)


2\Big(\sin^2(x)+\cos^2(x)-\sin(x)\cos(x)\Big)\Big(\sin(x) + \cos(x)\Big) = \\  2\Big(\sin^3(x) + cos^3(x)\Big)  \\  \\ 
2\Big(\sin^2(x)-\sin(x)\cos(x)+\cos^2(x)\Big)\Big(\sin(x) + \cos(x)\Big) = \\  2\Big(\sin^3(x) + cos^3(x)\Big)  \\  \\ 
\boxed{2\Big(\sin^3(x) + cos^3(x)\Big)  = 2\Big(\sin^3(x) + cos^3(x)\Big)  }



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