Question

Cos9xcos7x - sin9xsin7x = - 1/2

Answer 1

CosA*cosB-sinA*sinB=cos(A+B) 
поэтому: 
cos(9x+7x)=-1/2 
cos16x=-1/2 
16x=2pi/3 или 16x=4pi/3 
x=pi/24 или x=pi/12

Answer 2

To solve the equation cos(9x)cos(7x) - sin(9x)sin(7x) = -1/2, we can make use of the trigonometric identity known as the product-to-sum identity.

The product-to-sum identity states that cos(A)cos(B) - sin(A)sin(B) = cos(A + B). Using this identity, we can rewrite the equation as follows:

cos(9x)cos(7x) - sin(9x)sin(7x) = -1/2 cos(9x + 7x) = -1/2 cos(16x) = -1/2

Now, we need to find the values of x that satisfy the equation. The cosine function takes on values between -1 and 1, so we are looking for values of 16x that make cos(16x) equal to -1/2.

To find the solutions, we can look at the unit circle or use the inverse cosine function. The unit circle shows that cos(16x) = -1/2 occurs at two different angles: π/3 and 5π/3. These angles correspond to values of x as follows:

16x = π/3 => x = π/48 16x = 5π/3 => x = 5π/48

Therefore, the solutions to the equation cos(9x)cos(7x) - sin(9x)sin(7x) = -1/2 are x = π/48 and x = 5π/48.