Question

Cos10xcos6x-cos^2 8x=0

Answer 1

Используем формулу "произведение косинусов":

$\cos(\alpha)\cdot\cos(\beta) = \frac{1}{2}\cdot(\cos(\alpha + \beta) + \cos(\alpha - \beta))$

$\cos(10x)\cdot\cos(6x) = \frac{1}{2}\cdot(\cos(16x) + \cos(4x))$

$\cos^2(8x) = \cos(8x)\cdot\cos(8x) = \frac{1}{2}\cdot(\cos(16x) + \cos(0)) =$

$= \frac{1}{2}\cdot(\cos(16x) + 1)$

Тогда исходное уравнение равносильно следующему

$\frac{1}{2}\cdot(\cos(16x) + \cos(4x)) - \frac{1}{2}\cdot(\cos(16x)+1) = 0$

$\cos(16x) + \cos(4x) - \cos(16x) - 1 = 0$

$\cos(4x) = 1$

$4x = 2\pi\cdot m$

$x = \frac{\pi\cdot m}{2}$, m∈Z.

Answer 2

To solve the equation cos(10x)cos(6x) - cos^2(8x) = 0, we can use trigonometric identities to simplify it and find the solutions. Let's break it down step by step:

cos(10x)cos(6x) - cos^2(8x) = 0

Using the identity cos^2(x) = 1 - sin^2(x), we can rewrite the equation as:

cos(10x)cos(6x) - (1 - sin^2(8x)) = 0

Expanding the equation further:

cos(10x)cos(6x) - 1 + sin^2(8x) = 0

Now, we can use the identity sin^2(x) = 1 - cos^2(x) to replace sin^2(8x):

cos(10x)cos(6x) - 1 + (1 - cos^2(8x)) = 0

cos(10x)cos(6x) - 1 + 1 - cos^2(8x) = 0

Simplifying the equation:

cos(10x)cos(6x) - cos^2(8x) = 0

cos(10x)cos(6x) = cos^2(8x)

Using the identity cos(2x) = 2cos^2(x) - 1, we can rewrite the equation:

cos(10x)cos(6x) = (2cos^2(8x) - 1)

Now, we have an equation with similar terms on both sides. We can solve it by considering two cases:

Case 1: cos(10x) = 2cos^2(8x) - 1

Case 2: cos(6x) = 2cos^2(8x) - 1

Let's solve each case separately:

Case 1: cos(10x) = 2cos^2(8x) - 1

cos(10x) = 2(1 - sin^2(8x)) - 1

cos(10x) = 2 - 2sin^2(8x) - 1

cos(10x) = 1 - 2sin^2(8x)

Now, we have an equation in terms of cos(10x) and sin(8x). We can use the identity cos(2A) = 1 - 2sin^2(A) to rewrite it as:

cos(10x) = cos(2(8x))

Now, we equate the angles:

10x = 2(8x)

10x = 16x

10x - 16x = 0

-6x = 0

x = 0

So, one solution is x = 0.

Case 2: cos(6x) = 2cos^2(8x) - 1

cos(6x) = 2(1 - sin^2(8x)) - 1

cos(6x) = 2 - 2sin^2(8x) - 1

cos(6x) = 1 - 2sin^2(8x)

Again, we have an equation in terms of cos(6x) and sin(8x). We can use the identity cos(2A) = 1 - 2sin^2(A) to rewrite it as:

cos(6x) = cos(2(8x))

Now, we equate the angles:

6x = 2(