To solve this equation, we can start by using the double angle identity for sine:
sin(2x) = 2sin(x)cos(x)
Substitute this into the given equation:
sin^2(x) - √3/3 * 2sin(x)cos(x) = cos^2(x)
Next, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite the equation:
1 - sin^2(x) - √3/3 * 2sin(x)cos(x) = 1 - cos^2(x)
Rearranging terms:
1 - 1 - sin^2(x) + sin^2(x) - √3/3 * 2sin(x)cos(x) = cos^2(x) - cos^2(x)
-√3/3 * 2sin(x)cos(x) = 0
Multiplying through by 3/2:
-√3 * sin(x)cos(x) = 0
Since the product sin(x)cos(x) can only be 0 when either sin(x) or cos(x) is 0, we have two cases:
1) sin(x) = 0 which implies x = n*π (where n is an integer)2) cos(x) = 0 which implies x = (2n+1)π/2 (where n is an integer)
Thus, the solutions to the given equation are x = n*π and x = (2n+1)π/2 for all integers n.
To solve this equation, we can start by using the double angle identity for sine:
sin(2x) = 2sin(x)cos(x)
Substitute this into the given equation:
sin^2(x) - √3/3 * 2sin(x)cos(x) = cos^2(x)
Next, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite the equation:
1 - sin^2(x) - √3/3 * 2sin(x)cos(x) = 1 - cos^2(x)
Rearranging terms:
1 - 1 - sin^2(x) + sin^2(x) - √3/3 * 2sin(x)cos(x) = cos^2(x) - cos^2(x)
-√3/3 * 2sin(x)cos(x) = 0
Multiplying through by 3/2:
-√3 * sin(x)cos(x) = 0
Since the product sin(x)cos(x) can only be 0 when either sin(x) or cos(x) is 0, we have two cases:
1) sin(x) = 0 which implies x = n*π (where n is an integer)
2) cos(x) = 0 which implies x = (2n+1)π/2 (where n is an integer)
Thus, the solutions to the given equation are x = n*π and x = (2n+1)π/2 for all integers n.