Find the expression for cos 3x in terms of cos x

To find the expression for cos 3x in terms of cos x, we can use the cosine triple angle formula. The formula states:

cos(3x) = 4cos³(x) – 3cos(x)

Here’s a step-by-step explanation of how we arrive at this expression:

1. First, we start with the identity for cos(A + B):

   cos(A + B) = cos A * cos B – sin A * sin B

2. For our case, we set A = 2x and B = x:

   cos(3x) = cos(2x + x) = cos(2x)cos(x) – sin(2x)sin(x)

3. Next, we need to find expressions for cos(2x) and sin(2x):

   • cos(2x) = 2cos²(x) – 1
   • sin(2x) = 2sin(x)cos(x)

4. Substituting these values back into our equation for cos(3x):

   cos(3x) = (2cos²(x) – 1)cos(x) – (2sin(x)cos(x))sin(x)

5. Now, we know that sin²(x) = 1 – cos²(x). Substituting this into our expression gives us:

   cos(3x) = (2cos²(x) – 1)cos(x) – 2(1 – cos²(x))cos(x)

6. Expanding this leads to:

   cos(3x) = 2cos³(x) – cos(x) – 2cos(x) + 2cos³(x)

   Combining like terms results in:

   cos(3x) = 4cos³(x) – 3cos(x)

Therefore, the expression for cos(3x) in terms of cos x is:

cos(3x) = 4cos³(x) – 3cos(x)