Respuesta :
Answer: A
Step-by-step explanation:
The equation cos²(x) - sin²(x) = √3/2 can be rewritten using the trigonometric identity cos²(x) - sin²(x) = cos(2x). Therefore, we have cos(2x) = √3/2.
To find all the solutions to this equation, we can refer to the unit circle or use the special values of the cosine function. The cosine function takes on the value of √3/2 at two specific angles: π/6 and 11π/6. These angles correspond to the points on the unit circle where the x-coordinate is √3/2.
Since the cosine function has a period of 2π, we can add any integer multiple of 2π to these angles to get additional solutions. Therefore, the solutions to the equation cos(2x) = √3/2 are:
x = π/6 + 2πn and x = 11π/6 + 2πn,
where n is an integer.
However, we need to remember that we are solving for x, not 2x. So, we divide each solution by 2:
x = π/6 + 2πn/2 and x = 11π/6 + 2πn/2.
Simplifying further:
x = π/6 + πn and x = 11π/6 + πn,
where n is an integer.
Therefore, the correct answer is:
A. x = π/6 + 2πn and x = 11π/6 + 2πn.
