To solve this trigonometric equation, we can use the double angle formula for cosine:
cos(2x) = 2cos^2(x) - 1
Substitute this formula into the equation:
(1 - cos(6x)) * (2cos^2(x) - 1) = sin^2(3x)
Expand the left side:
2cos^2(x) - cos(6x) - 2cos^2(x) + 1 = sin^2(3x)
Simplify:
Since sin^2(x) = 1 - cos^2(x), we can rewrite the equation as:
cos(6x) = cos^2(3x)
Apply the double angle formula for cosine again to the right side:
cos(6x) = (1 + cos(6x))/2
Rearranging the equation gives:
2cos(6x) = 1 + cos(6x)
cos(6x) = 1
As a result, this equation simplifies to:
Since the range of cosine is -1 to 1, the only solution to this equation is cos(6x) = 1, which implies that 6x must be a multiple of 2π:
6x = 2πn, where n is any integer
x = πn/3, where n is any integer
Therefore, the general solution to the equation is x = πn/3, where n is an integer.
To solve this trigonometric equation, we can use the double angle formula for cosine:
cos(2x) = 2cos^2(x) - 1
Substitute this formula into the equation:
(1 - cos(6x)) * (2cos^2(x) - 1) = sin^2(3x)
Expand the left side:
2cos^2(x) - cos(6x) - 2cos^2(x) + 1 = sin^2(3x)
Simplify:
cos(6x) + 1 = sin^2(3x)Since sin^2(x) = 1 - cos^2(x), we can rewrite the equation as:
cos(6x) + 1 = 1 - cos^2(3x)cos(6x) = cos^2(3x)
Apply the double angle formula for cosine again to the right side:
cos(6x) = (1 + cos(6x))/2
Rearranging the equation gives:
2cos(6x) = 1 + cos(6x)
cos(6x) = 1
As a result, this equation simplifies to:
cos(6x) = 1
Since the range of cosine is -1 to 1, the only solution to this equation is cos(6x) = 1, which implies that 6x must be a multiple of 2π:
6x = 2πn, where n is any integer
x = πn/3, where n is any integer
Therefore, the general solution to the equation is x = πn/3, where n is an integer.