Prove or disprove the following identity:
(sin(10x))/(sin(x)+sin(9x)) = (cos(5x))/(cos4x))
To prove or disprove the given identity, we will start by working with the left side of the equation and attempting to simplify it to match the right side.
Step 1: Convert the numerator on the left side to use the double-angle identity for sine:
sin(10x) = 2*sin(5x)*cos(5x)
Now we have:
(2*sin(5x)*cos(5x))/(sin(x) + sin(9x))
Step 2: Use the sum-to-product identity for cosine to simplify the denominator:
sin(x) + sin(9x) = 2*sin(5x)*cos(4x)
Now we have:
(2*sin(5x)*cos(5x))/(2*sin(5x)*cos(4x))
Step 3: Cancel out common factors:
sin(5x) cancels out in the numerator and denominator, leaving us with:
cos(5x)/(cos(4x))
This matches the right side of the equation, so we have successfully proven the given identity:
(sin(10x))/(sin(x) + sin(9x)) = (cos(5x))/(cos(4x))
To prove or disprove this identity, we can simplify both sides of the equation and check if they are equal. Let's start with the left side:
(sin(10x))/(sin(x) + sin(9x))
Using the sum-to-product formula, we can rewrite the denominator as a product:
(sin(10x))/(2sin((x + 9x)/2)cos((x - 9x)/2))
Simplifying, we get:
(sin(10x))/(2sin(5x)cos(-4x))
Now, notice that cos(-4x) is equal to cos(4x) since cosine is an even function. Continuing with the simplification:
(sin(10x))/(2sin(5x)cos(4x))
Now let's simplify the right side of the equation:
(cos(5x))/(cos(4x))
Given the simplified form of both sides of the equation, we can see that the two sides match. Therefore, we know that the given identity is true.
Keep in mind that when proving or disproving identities, simplifying both sides and showing their equivalence is the general method.
sin(x)+sin(9x) = 2sin(5x)cos(4x), so
sin(10x)/(2sin(5x)cos(4x)) = cos(5x)/cos(4x)
sin(10x)cos(4x) = 2cos(5x)sin(5x)cos(4x)
sin(10x)cos(4x) = sin(10x)cos(4x)
QED