Question

Cos^2(2x)-sin^2(2x)=-1

Answer 1

Воспользуемся основным тригонометрическим тождеством: sin^2(t) + cos^2(t) = 1. Выразим из него синус квадрат: sin^2(t) = 1 - cos^2(t). Таким образом, исходное уравнение запишется так:
cos^2(2x) - 1 + cos^2(2x) = -1;
2cos^2(2x) = 0;
cos^2(2x) = 0;
cos(2x) = 0;
2x = π/2 + πn, n∈Z;
x = π/4 + πn/2, n∈Z.
Ответ: 

Answer 2

Let's work on simplifying the expression on the left-hand side and see if it indeed equals -1.

We'll start with the given expression: cos^2(2x) - sin^2(2x)

Now, we can use trigonometric identities to simplify this expression. One of the well-known trigonometric identities is:

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

We can use this identity to simplify the given expression:

cos^2(2x) - sin^2(2x) = cos(2 * 2x) = cos(4x)

So, the simplified expression is cos(4x).

Now, let's evaluate cos(4x) to check if it equals -1:

cos(4x) = -1

Since cos(4x) = -1, the original expression cos^2(2x) - sin^2(2x) does NOT equal -1. Instead, it simplifies to -1. The equation is true:

cos^2(2x) - sin^2(2x) = -1.