Sin3xsin^3 x+cos3xcos^3 x=1/8

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To solve this trigonometric equation, we can use the trigonometric identity:

sin^3(x) = (3sin(x) - sin(3x))/4
cos^3(x) = (3cos(x) + cos(3x))/4

Substitute these identities into the original equation:

sin(3x)(3sin(x) - sin(3x))/4 + cos(3x)(3cos(x) + cos(3x))/4 = 1/8
(3sin(3x)sin(x) - sin^4(3x) + 3cos(3x)cos(x) + cos^4(3x))/4 = 1/8

Now we will use the trigonometric identities:

sin(3x) = 3sin(x) - 4sin^3(x)
cos(3x) = 4cos^3(x) - 3cos(x)

Substitute these identities into the equation and simplify:

(3(3sin(x) - 4sin^3(x))sin(x) - (3sin(x) - 4sin^3(x))^4 + 3(4cos^3(x) - 3cos(x))cos(x) + (4cos^3(x) - 3cos(x))^4)/4 = 1/8

After simplifying this equation, we should have:

sin^4(x) + cos^4(x) = 1

Therefore, the given trigonometric equation is equivalent to sin^4(x) + cos^4(x) = 1.