Question

Sa se arate ca expresia E(x) = $sin^8x - cos^8x + 4cos^6x - 6cos^4x + 4cos^2x$ nu depinde de X.

Answer 1

   
[tex]\text{Folosim formula:}\\ \sin^2x+ \cos^2x = 1 ~~~\Longrightarrow~~~ \sin^2x=1-\cos^2x\\\\ \text{Rezolvare:}\\ \text{Ne ocupam de }~~~\sin^8x\\\\ \sin^8x = \Big(\sin^2 x\Big)^4 = \Big(1-\cos^2x\Big)^4= \left[\Big(1-\cos^2x\Big)^2\right]^2=\\\\\\ = \left[1-2\cos^2x + \cos^4x\right]^2=\\\\=(1-2\cos^2x + \cos^4x)(1-2\cos^2x + \cos^4x)=\\\\ = 1-4\cos^2x + \underline{2\cos^4x} +\underline{4\cos^4x} -4\cos^6x+\cos^8x = \\\\ = \boxed{1-4\cos^2x + 6\cos^4x -4\cos^6x+\cos^8x }[/tex]


[tex]\text{Acum ne ocupam de intreaga formula:}\\\\ \sin^8x - \cos^8x + 4\cos^6x - 6\cos^4x + 4\cos^2x =\\\\ =1- \underline{4\cos^2x} + \underline{\underline{6\cos^4x}} -\underset{=====}{4\cos^6x} +\underset{******}{\cos^8x} -\\ -\underset{******}{\cos^8x} + \underset{=====}{4\cos^6x} - \underline{\underline{6\cos^4x}} + \underline{4\cos^2x} = \boxed{\boxed{1}}\\\\ \boxed{1 \in N \text{ si nu depinde de }x.}[/tex]