Question

Cos^2 x/6 - sin^2 x/6 = 1/2

Answer 1

В выражении, стоящей слева в уравнении, применим косинус двойного угла

                                       $\cos \left(2\cdot \frac{x}{6}\right)=\frac{1}{2}\\ \\ \cos \frac{x}{3}=\frac{1}{2}\\ \\ \frac{x}{3}=\pm \frac{\pi}{3}+2\pi n,n \in \mathbb{Z}\\ \\ x=\pm \pi +6\pi n,n \in \mathbb{Z}$

Answer 2

To solve the equation cos^2(x)/6 - sin^2(x)/6 = 1/2, we can simplify it and solve for x. Let's start by multiplying both sides of the equation by 6 to eliminate the denominators:

6 * (cos^2(x)/6) - 6 * (sin^2(x)/6) = 6 * (1/2).

This simplifies to:

cos^2(x) - sin^2(x) = 3.

Now, we can use the trigonometric identity cos^2(x) - sin^2(x) = cos(2x) to further simplify the equation:

cos(2x) = 3.

To find the values of x, we need to solve for 2x:

2x = arccos(3).

However, the value of arccos(3) is not defined since the range of the arccosine function is -1 to 1. Therefore, there are no real solutions to the given equation.