If $a^{2} + b^{2} = c^{2}$ , then show that $a^{6} + b^{6} + 3 a^{2} b^{2} c^{2} = c^{6}$

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Solution

Accordingto question,

$a^{2} + b^{2} = c^{2}$ ....(1)

By cubing both side,

$\Rightarrow$ $(a^{2} + b^{2})^{3} = (c^{2})^{3}$

$\Rightarrow$ $a^{6} + b^{6} + 3 (a^{2} + b^{2}) a^{2} b^{2} = c^{6}$

By using (1)

$\Rightarrow$ $a^{6} + b^{6} + 3 (c^{2}) a^{2} b^{2} = c^{6}$

$\Rightarrow$ $a^{6} + b^{6} + 3 a^{2} b^{2} c^{2} = c^{6}$

Hence proved

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