Establish the trigonometric identity (Hint: Use sum to product)
cos(5x)+cos(3x)/sin(5x)-sin(3x)=cot(x)
To establish the trigonometric identity cos(5x) + cos(3x) / sin(5x) - sin(3x) = cot(x), we can use the sum-to-product identity for cosine.
The sum-to-product identity states that for any two angles A and B:
cos(A) + cos(B) = 2 * cos((A + B) / 2) * cos((A - B) / 2)
To use this identity, let A = 5x and B = 3x:
cos(5x) + cos(3x) = 2 * cos((5x + 3x) / 2) * cos((5x - 3x) / 2)
cos(5x) + cos(3x) = 2 * cos(4x) * cos(x)
Now, let's apply the sum-to-product identity for sine:
sin(A) - sin(B) = 2 * sin((A - B) / 2) * cos((A + B) / 2)
Again, let A = 5x and B = 3x:
sin(5x) - sin(3x) = 2 * sin((5x - 3x) / 2) * cos((5x + 3x) / 2)
sin(5x) - sin(3x) = 2 * sin(x) * cos(4x)
Now we can substitute the results into the given equation:
(cos(5x) + cos(3x)) / (sin(5x) - sin(3x)) = cot(x)
(2 * cos(4x) * cos(x)) / (2 * sin(x) * cos(4x)) = cot(x)
The 2's cancel out:
cos(x) / sin(x) = cot(x)
Since cot(x) = cos(x) / sin(x), the identity is established.