Sinx - cosx = v2sin6x

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To solve the equation sinx - cosx = sqrt(2)sin(6x), we can rewrite the right side using the angle addition formula for sine.

sinx - cosx = sqrt(2)(sinxcos(6x) + cosxsin(6x))

Expanding the right side further:

sinx - cosx = sqrt(2)(sinxcos(6x) + cosxsin(6x))
= sqrt(2)(sinxcosxcos5x + cosxsinxcos5x)
= sqrt(2)(sinxcosxcos5x + sinxcosxcos(π/2-5x))
= sqrt(2)(sinxcosx(cos5x + cos(π/2-5x)))
= sqrt(2)(sinxcosx(2cos(π/4)cos(5x-π/4)))
= sqrt(2)(sinxcosx(cos(5x-π/4)))
= sqrt(2)sinxcosx*cos(5x-π/4)

Now, we have:

sinx - cosx = sqrt(2)sinxcosxcos(5x-π/4)

Since sinx - cosx = sqrt(2)sinxcosxcos(5x-π/4), we can divide both sides by sinx*cosx to get:

1 = sqrt(2)cos(5x-π/4)

Now, isolating cos(5x-π/4):

cos(5x-π/4) = 1/sqrt(2)
5x - π/4 = ± π/4 + 2πn

Solving for x:

5x = π/2, 3π/2, π/4 + 2πn, 5π/4 + 2πn
x = π/10, 3π/10, π/20 + 2πn, 5π/20 + 2πn

Therefore, the solutions to the equation sinx - cosx = sqrt(2)sin(6x) are x = π/10, 3π/10, π/20 + 2πn, and 5π/20 + 2πn, where n is an integer.