Solve the equation sin^6x + cos^6x = 1 .

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll apply the Pythagorean identity to solve the equation:

(sin x)^2 + (cos x)^2 = 1

We'll write (sin x)^6 + (cos x)^6 = ((sin x)^2 + (cos x)^2)^3 - 3(sin x)^2*(cos x)^2)((sin x)^2 + (cos x)^2)

We'll replace the sum (sin x)^2 + (cos x)^2 by 1 and we'll get:

(sin x)^6 + (cos x)^6 = 1^3 - 3*1*[(sin x)^2*(cos x)^2]

But (sin x)^6 + (cos x)^6 =1

We'll re-write the equation:

1 = 1 - 3*[(sin x)^2*(cos x)^2]

We'll eliminate like terms and we'll get:

3*[(sin x)^2*(cos x)^2] = 0

We'll divide by 3:

[(sin x)^2*(cos x)^2] = 0

We'll cancel each factor:

(sin x)^2 = 0

sin x = 0 => x = arcsin 0 + k*pi => x = 0 or x = pi or x = 2pi

(cos x)^2 = 0 => cos x = 0 => x = pi/2 or x = 3pi/2

The complete set of solutions of the equation are: {0 , pi/2 , pi, 3pi/2 , 2pi}.

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