# Trigonometric identities

Trigonometry equalities, inequalities and expressions - sin, cos, tan, cot

### Trigonometric identities

Prove the trigonometric identity:
$$3(sin x-cos x)^4+4(sin^6 x+cos^6 x)+6(sin x+cos x)^2=13$$
I am puzzled with the identity solution........
Rachel

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To solve the problem we have to use the major trigonometric identity:
$$sin^2x + cos^2x = 1$$
and short multiplication formulas.
$$(sin-cos)^4 = (sin^2x - 2sinxcosx + cos^2x)^2 = (1 - 2sinxcosx)^2 = 1 - 4sinxcosx + 4sin^2xcos^2x$$

$$(sin^6x + cos^6x) = (sin^2x)^3 + (cos^2x)^3 =$$
$$= (sin^2x + cos^2x)(sin^4x + sin^2xcos^2x + cos^4x) =$$
$$=1.(((sin^2x)^2+ 2sin^2xcos^2x + (cos^2x)^2) - sin^2xcos^2x)=$$
$$=((sin^2x + cos^2x)^2 - sin^2xcos^2x) = (1 - sin^2xcos^2x)$$

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