Question

Prove it :

i]

2{sin^6(x) + cos^6(x)}+1=3{sin^4(x)+cos^4x}
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Answer 1

To Prove:

$\sf 2({sin}^{6}x + {cos}^{6}x ) +1 = 3({sin}^{4}x + {cos}^{4}x)$

Proof:

LHS:

$\sf \implies 2({sin}^{6}x + {cos}^{6}x ) +1$

$\sf \implies 2[{({sin}^{2}x + {cos}^{2}x )}^{3} - 3{sin}^{2}x \times {cos}^{2}({sin}^{2}x + {cos}^{2})] +1$

$\sf \implies 2[{1}^{3} - 3{sin}^{2}x \times {cos}^{2}x(1)] +1$

$\sf \implies 2[1- 3{sin}^{2}x \times {cos}^{2}x] +1$

$\sf \implies 2- 6{sin}^{2}x \times {cos}^{2}x+1$

$\sf \implies 3 - 6{sin}^{2}x \times {cos}^{2}x$

$\sf \implies 3(1 -2 {sin}^{2}x \times {cos}^{2}x)$

$\sf \implies 3[{({sin}^{2}x + {cos}^{2}x)}^{2}-2 {sin}^{2}x \times {cos}^{2}x]$

$\sf \implies 3({sin}^{4}x + {cos}^{4}x + \cancel{2 {sin}^{2}x \times {cos}^{2}x}-\cancel{2 {sin}^{2}x \times {cos}^{2}x}$

$\sf \implies 3({sin}^{4}x + {cos}^{4}x)$

RHS:

$\sf \implies 3({sin}^{4}x + {cos}^{4}x)$

Hence Proved!

Answer 2

Answer:

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