Question

What are all the solutions to the equation cos��(x) - sin��(x) = ���(2)/2?

1) x = \frac{\pi}{8} + \pi n$ and $x = \frac{7\pi}{8} + \pi n
2) x = \frac{3\pi}{8} + \pi n$ and $x = \frac{5\pi}{8} + \pi n
3) x = \frac{\pi}{8} + 2\pi n$ and $x = \frac{7\pi}{8} + 2\pi n
4) x = \frac{3\pi}{8} + 2\pi n$ and $x = \frac{5\pi}{8} + 2\pi n

Answer 1

The solutions to the equation cos²(x) - sin²(x) = √2/2 are x = π/6 + 2πn and x = 11π/6 + 2πn, as well as x = 5π/6 + 2πn and x = 7π/6 + 2πn.

Step-by-step explanation:

To find the solutions to the equation cos²(x) - sin²(x) = √2/2, we need to use the trigonometric identity cos²(x) - sin²(x) = 1. By substituting √2/2 for the right side of the equation, we get cos²(x) - sin²(x) = 1/2. Simplifying further, we have cos²(x) - (1 - cos²(x)) = 1/2. This simplifies to 2cos²(x) - 1 = 1/2, which leads to 2cos²(x) = 3/2.

Dividing both sides by 2, we get cos²(x) = 3/4. Taking the square root of both sides, we have cos(x) = ±√(3/4). Evaluating the square root, we find cos(x) = ±√3/2.

Since cos(x) is equal to √3/2 at x = π/6 and x = 11π/6, and cos(x) is equal to -√3/2 at x = 5π/6 and x = 7π/6, the solutions to the equation are:

• x = π/6 + 2πn and x = 11π/6 + 2πn
• x = 5π/6 + 2πn and x = 7π/6 + 2πn