Sin(7П+x)=cos(9П+2x)

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To solve this equation, we need to use the following trigonometric identities:

sin(π + x) = -sin(x)
cos(π + x) = -cos(x)

With these identities in mind, we can rewrite the given equation as:

-sin(x) = cos(2x)

Now we can use the double angle formula for cosine to rewrite cos(2x) in terms of cos(x):

cos(2x) = 2cos^2(x) - 1

Substitute this into the equation:

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

Now we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite cos^2(x) in terms of sin(x):

-sin(x) = 2(1 - sin^2(x)) - 1
-sin(x) = 1 - 2sin^2(x) - 1
0 = -2sin^2(x)
0 = sin^2(x)

Therefore, the solution to the equation sin(7π + x) = cos(9π + 2x) is sin(x) = 0 or x = nπ, where n is an integer.