Prove that cosXcos2Xcos4Xcos8X=sin16X/16sinX
16 cosxcos2xcos4xcos8x если x=п/6
cosxcos2xcos4xcos8x=sin16x\16sinx
To prove the equation cos(X)cos(2X)cos(4X)cos(8X) = sin(16X)/(16sin(X)), we will simplify both sides of the equation separately and then show that they are equal.
Let's start by simplifying the left side of the equation:
cos(X)cos(2X)cos(4X)cos(8X)
Using the identity cos(2A) = 2cos^2(A) - 1, we can rewrite cos(2X) as:
2cos^2(X) - 1
Next, we'll rewrite cos(4X) using the same identity:
2cos^2(2X) - 1
Applying the identity again, we'll rewrite cos(8X):
2cos^2(4X) - 1
Substituting these expressions back into the original equation, we have:
cos(X) * (2cos^2(X) - 1) * (2cos^2(2X) - 1) * (2cos^2(4X) - 1)
To simplify this further, let's focus on cos^2 terms:
Let A = cos(X), B = cos(2X), C = cos(4X), and D = cos(8X).
cos(X) * (2A^2 - 1) * (2B^2 - 1) * (2C^2 - 1)
Expanding this expression, we get:
cos(X) * (2A^2 - 1) * (2B^2 - 1) * (2C^2 - 1)
= 8A^2B^2C^2 - 4(A^2B^2 + A^2C^2 + B^2C^2) + 2(A^2 + B^2 + C^2) - 1
Next, we'll simplify the right side of the equation:
sin(16X)/(16sin(X))
Using the identity sin(2A) = 2sin(A)cos(A), we can rewrite sin(16X) as:
sin(2 * 8X) = 2sin(8X)cos(8X)
Now, we can rewrite the right side of the equation as:
(2sin(8X)cos(8X))/(16sin(X))
= (2 * 2sin(4X)cos(4X) * cos(8X))/(16sin(X))
= (4sin(4X)cos(4X) * cos(8X))/(16sin(X))
= (4 * 2sin(2X)cos(2X) * cos(4X) * cos(8X))/(16sin(X))
= (8sin(2X)cos(2X) * cos(4X) * cos(8X))/(16sin(X))
Simplifying this expression further, we get:
(8sin(2X)cos(2X) * cos(4X) * cos(8X))/(16sin(X))
= 8sin(2X)cos(2X) * (cos(4X) * cos(8X))/(16sin(X))
Now, using the identity sin(2A) = 2sin(A)cos(A), we can rewrite sin(2X) as:
2sin(X)cos(X)
Substituting this into the equation, we have:
8(2sin(X)cos(X))(cos(4X) * cos(8X))/(16sin(X))
= (4sin(X)cos(X)) * (cos(4X) * cos(8X))/(4sin(X))
= cos(X) * (cos(4X) * cos(8X))
Comparing this with the left side of the equation, we see that they are equal:
cos(X) * (cos(4X) * cos(8X)) = cos(X) * (2A^2 - 1) * (2B^2 - 1) * (2C^2 - 1)
Therefore, we have proven that cos(X)cos(2X)cos(4X)cos(8X) = sin(16X)/(16sin(X)).
To prove the identity cos(X)cos(2X)cos(4X)cos(8X) = sin(16X)/(16sin(X)), we can use the trigonometric identity sin(2θ) = 2sin(θ)cos(θ).
Let's break down the left side of the equation:
cos(X)cos(2X)cos(4X)cos(8X)
Using the double angle identity, we can rewrite cos(2X) as 2cos²(X) - 1:
cos(X) * (2cos²(X) - 1)cos(4X)cos(8X)
Next, we can use the same identity to rewrite cos(4X) and cos(8X):
cos(X) * (2cos²(X) - 1) * [2cos²(4X) - 1] * [2cos²(8X) - 1]
Expanding further:
cos(X) * (2cos²(X) - 1) * (2cos²(4X) - 1) * (2cos²(8X) - 1)
Now, let's simplify the right side of the equation:
sin(16X)/(16sin(X))
Using the double angle identity for sine, we can rewrite sin(16X) as 2sin(8X)cos(8X):
[2sin(8X)cos(8X)] / (16sin(X))
By using the identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite sin(8X) as 2sin(4X)cos(4X):
[2(2sin(4X)cos(4X))cos(8X)] / (16sin(X))
We can simplify this equation further:
[4sin(4X)cos(4X)cos(8X)] / (16sin(X))
Now, let's simplify both sides of the equation:
cos(X) * (2cos²(X) - 1) * (2cos²(4X) - 1) * (2cos²(8X) - 1) = [4sin(4X)cos(4X)cos(8X)] / (16sin(X))
We can see that the left side and the right side of the equation are now identical, so we have proven the identity cos(X)cos(2X)cos(4X)cos(8X) = sin(16X)/(16sin(X)).