Question

Cos3x*cos6x=cos4x*cos7x

Answer 1

Используем формулу:cosa*cosb=(cos(a-b)+ cos(a+b))/2.
Имеем: cos3x*cos6x=(cos(3х-6х)+ cos(3х+6х))/2=(cos(3х)+ cos(9х))/2;
cos4x*cos7x=(cos(4х-7х)+ cos(4х+7х))/2=(cos(3х)+ cos(11х))/2;
значит, (cos(3х)+ cos(9х))/2=(cos(3х)+ cos(11х))/2.Отсюда, cos(9х))/2= =cos(11х))/2, cos(9х) =cos(11х),9х=11х, х=0.
Ответ: 0

Answer 2

To solve the equation cos(3x) * cos(6x) = cos(4x) * cos(7x), you can use trigonometric identities and algebraic manipulations. Here's how you can solve it step by step:

1. Start with the equation: cos(3x) * cos(6x) = cos(4x) * cos(7x)

2. Use the trigonometric identity for the product of two cosines: cos(A) * cos(B) = 0.5 * [cos(A + B) + cos(A - B)]

3. Apply the identity to the left side of the equation: 0.5 * [cos(3x + 6x) + cos(3x - 6x)] = cos(4x) * cos(7x)

4. Simplify the expressions inside the brackets: 0.5 * [cos(9x) + cos(-3x)] = cos(4x) * cos(7x)

5. Since the cosine function is an even function (cos(-x) = cos(x)), you can simplify further: 0.5 * [cos(9x) + cos(3x)] = cos(4x) * cos(7x)

6. Now, use the product-to-sum identity for cosines: cos(A) * cos(B) = 0.5 * [cos(A + B) + cos(A - B)]

   Apply this identity to the right side of the equation: 0.5 * [cos(4x + 7x) + cos(4x - 7x)] = 0.5 * [cos(11x) + cos(-3x)]

7. Simplify the right side of the equation using the even property of cosine: 0.5 * [cos(11x) + cos(3x)] = 0.5 * [cos(11x) + cos(3x)]

8. Now, the equation is simplified to: 0.5 * [cos(9x) + cos(3x)] = 0.5 * [cos(11x) + cos(3x)]

9. Since both sides of the equation are now identical, the equation is satisfied for all values of x. In other words, the equation is an identity, and it holds true for any value of x.

So, the solution is that cos(3x) * cos(6x) = cos(4x) * cos(7x) is an identity, and it is true for all values of x.