Show thet: ( ^6x+ ^6x)-3( ^4x+ ^4x)+1=0 - Vidyadip

Question

Show thet:
$(\sin^6\text{x}+\cos^6\text{x})-3(\sin^4\text{x}+\cos^4\text{x})+1=0$

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Answer

$\text{LHS}=2(\sin^6\text{x}+\cos^6\text{x})-3(\sin^4\text{x}+\cos^4\text{x})+1$
$=2\big((\sin^2)^3+(\cos^2\text{x})^3\big)-3(\sin^4\text{x}+\cos^4\text{x})+1$ $\big[\because\text{a}^3+\text{b}^3=(\text{a+b})\big(\text{a}^2\text{ab+b}^2\big)\big]$
$=2[(\sin^2\text{x}+\cos^4\text{x})(\sin^4\text{x}-\sin^2\text{x}\cos^2\text{x}+\cos^4\text{x})]-3(\sin^4\text{x}+\cos^4\text{x})+1$
$=-[\sin^4\text{x}+\cos^4\text{x}+2\sin^2\text{x}\cos^2\text{x}]+1$
$=-[\sin^2\text{x}+\cos^2\text{x}]+1$
$=-1+1$
$=0=\text{RHS}$

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