It appears that the original equation cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x) does not simplify to cos(7x)cos(2x). The two sides are not equal. Therefore, the given trigonometric identity is not valid.
To verify the trigonometric identity: cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x),We'll use trigonometric identities and simplifications ...
To verify the trigonometric identity: cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x),We'll use trigonometric identities and simplifications ...To verify the trigonometric identity: cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x),We'll use trigonometric identities and simplifications
Answers & Comments
To verify the trigonometric identity: cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x),
We'll use trigonometric identities and simplifications step by step:
1. We'll start with the left-hand side (LHS) of the equation:
LHS = cos(4x)cos(x) - sin(6x)sin(3x)
2. We'll use the double-angle identity for cosine: cos(2θ) = 2cos²(θ) - 1.
Apply this identity to cos(4x):
cos(4x) = 2cos²(2x) - 1
3. Expand sin(6x)sin(3x) using the product-to-sum identities:
sin(6x)sin(3x) = 1/2[cos(3x - 6x) - cos(3x + 6x)]
sin(6x)sin(3x) = 1/2[cos(-3x) - cos(9x)]
4. Now, we'll use the double-angle identity for cosine on cos(9x):
cos(9x) = 2cos²(4.5x) - 1
cos(9x) = 2cos²(2x) - 1
5. Plug these results back into the original equation:
LHS = (2cos²(2x) - 1)cos(x) - (1/2)[cos(-3x) - 2cos²(2x) + 1]
6. Simplify LHS further:
LHS = 2cos³(2x) - cos(x) - (1/2)[cos(3x) - 2cos²(2x) - 1]
7. Use the double-angle identity for cosine again on cos(2x):
cos(2x) = 2cos²(x) - 1
8. Substitute cos(2x) into LHS:
LHS = 2cos³(2x) - cos(x) - (1/2)[cos(3x) - 2(2cos²(x) - 1) - 1]
9. Expand and simplify:
LHS = 2cos³(2x) - cos(x) - (cos(3x) - 4cos²(x) + 1) - 1/2
10. Distribute the 1/2 to each term in the square brackets:
LHS = 2cos³(2x) - cos(x) - cos(3x) + 4cos²(x) - 1 - 1/2
11. Combine like terms:
LHS = 2cos³(2x) + 4cos²(x) - cos(x) - cos(3x) - 3/2
Now, let's simplify the right-hand side (RHS):
RHS = cos(7x)cos(2x)
12. Apply the double-angle identity for cosine to cos(7x):
cos(7x) = 2cos²(3.5x) - 1
cos(7x) = 2cos²(2x) - 1
13. Substitute cos(7x) into RHS:
RHS = (2cos²(2x) - 1)cos(2x)
14. Distribute cos(2x) to the terms in the brackets:
RHS = 2cos²(2x)cos(2x) - cos(2x)
Now, let's compare the simplified LHS and RHS:
LHS = 2cos³(2x) + 4cos²(x) - cos(x) - cos(3x) - 3/2
RHS = 2cos²(2x)cos(2x) - cos(2x)
It appears that the original equation cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x) does not simplify to cos(7x)cos(2x). The two sides are not equal. Therefore, the given trigonometric identity is not valid.
To verify the trigonometric identity: cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x),We'll use trigonometric identities and simplifications ...
To verify the trigonometric identity: cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x),We'll use trigonometric identities and simplifications ...To verify the trigonometric identity: cos(4x)cos(x) - sin(6x)sin(3x) = cos(7x)cos(2x),We'll use trigonometric identities and simplifications