Trigonometric identities

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

Trigonometric identities

Postby Rachel » Tue Sep 07, 2010 3:24 am

Prove the trigonometric identity:
[tex]3(sin x-cos x)^4+4(sin^6 x+cos^6 x)+6(sin x+cos x)^2=13[/tex]
I am puzzled with the identity solution........
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Postby Math Tutor » Tue Sep 07, 2010 4:45 am

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

[tex](sin^6x + cos^6x) = (sin^2x)^3 + (cos^2x)^3 =[/tex]
[tex]= (sin^2x + cos^2x)(sin^4x + sin^2xcos^2x + cos^4x) =[/tex]
[tex]=1.(((sin^2x)^2+ 2sin^2xcos^2x + (cos^2x)^2) - sin^2xcos^2x)=[/tex]
[tex]=((sin^2x + cos^2x)^2 - sin^2xcos^2x) = (1 - sin^2xcos^2x)[/tex]

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