Question

Cos(3x+п/3)-cos3x=cos(x+п/3)-cosx​

Answer 1

To solve the equation cos(3x + π/3) - cos(3x) = cos(x + π/3) - cos(x), we can simplify the expression using trigonometric identities.

Step 1: Use the identity cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2):

-2sin((3x + π/3 + 3x)/2)sin((3x + π/3 - 3x)/2) = -2sin((x + π/3 + x)/2)sin((x + π/3 - x)/2)

Simplifying further:

-2sin((6x + π/3)/2)sin(π/6) = -2sin((2x + π/3)/2)sin(π/6)

Step 2: Use the identity sin(π/6) = 1/2:

-2sin((6x + π/3)/2) * (1/2) = -2sin((2x + π/3)/2) * (1/2)

Step 3: Cancel out the factor of -2 and multiply both sides by 2 to get rid of the denominators:

sin((6x + π/3)/2) = sin((2x + π/3)/2)

Step 4: Use the identity sin(A) = sin(B) implies A = B + 2πn or A = π - B + 2πn:

(6x + π/3)/2 = (2x + π/3)/2 + 2πn

Simplifying:

3x + π/3 = x + π/3 + 4πn

Step 5: Subtract (x + π/3) from both sides:

2x = 4πn

Step 6: Divide by 2:

x = 2πn

Therefore, the solutions to the equation cos(3x + π/3) - cos(3x) = cos(x + π/3) - cos(x) are x = 2πn, where n is an integer.