prove that
sin8x = 8sinx cosx cos2x cos4x
sin8x = 2 sin4x cos4x
sin4x = 2 sin2x cos2x
and so on
To prove that sin(8x) = 8sin(x)cos(x)cos(2x)cos(4x), we will make use of trigonometric identities and properties.
Step 1: Start with the double angle formula for cosine: cos(2θ) = 2cos^2(θ) - 1
Step 2: Apply the double angle formula to cos(4x) and cos(2x) in the equation sin(8x) = 8sin(x)cos(x)cos(2x)cos(4x):
sin(8x) = 8sin(x)cos(x)cos(2x)(2cos^2(4x) - 1)
Step 3: Apply the double angle formula again to cos(4x) inside the expression:
sin(8x) = 8sin(x)cos(x)cos(2x)(2[2cos^2(2x) - 1]^2 - 1)
Step 4: Simplify the expression inside parentheses:
sin(8x) = 8sin(x)cos(x)cos(2x)(2[4cos^4(2x) - 4cos^2(2x) + 1] - 1)
Step 5: Multiply through by 2:
sin(8x) = 8sin(x)cos(x)cos(2x)(8cos^4(2x) - 8cos^2(2x) + 2 - 1)
Step 6: Simplify further:
sin(8x) = 8sin(x)cos(x)cos(2x)(8cos^4(2x) - 8cos^2(2x) + 1)
Step 7: Factor out common terms:
sin(8x) = 8sin(x)cos(x)cos(2x)[(2cos^2(2x) - 1)^2]
Step 8: Apply the double angle formula for cosine again to (2cos^2(2x) - 1) within the brackets:
sin(8x) = 8sin(x)cos(x)cos(2x)[(2[2cos^2(x) - 1] - 1)^2]
Step 9: Simplify the expression within the brackets:
sin(8x) = 8sin(x)cos(x)cos(2x)[(4cos^2(x) - 3)^2]
Step 10: Simplify further:
sin(8x) = 8sin(x)cos(x)cos(2x)(16cos^4(x) - 24cos^2(x) + 9)
Step 11: Expand and distribute:
sin(8x) = 8sin(x)cos(x)[16cos^5(x) - 24cos^3(x) + 9cos(2x)]
Step 12: Apply the double angle formula for sine: sin(2θ) = 2sin(θ)cos(θ)
sin(8x) = 8sin(x)cos(x)[16cos^5(x) - 24cos^3(x) + 9(2sin(x)cos(x))]
Step 13: Simplify:
sin(8x) = 8sin(x)cos(x)[16cos^5(x) - 24cos^3(x) + 18sin(x)cos(x)]
Step 14: Factor out common terms:
sin(8x) = 8sin(x)cos(x)(16cos^5(x) - 24cos^3(x) + 18sin(x)cos(x))
Step 15: Simplify further:
sin(8x) = 8sin(x)cos(x)(2sin(x)cos(x)(8cos^4(x) - 12cos^2(x) + 9))
Step 16: Simplify the expression within the parentheses:
sin(8x) = 8sin(x)cos(x)(2sin(x)cos(x)(2cos^2(x) - 3)^2)
Step 17: Apply the double angle formula for cosine again:
sin(8x) = 8sin(x)cos(x)(2sin(x)cos(x)(2[2cos^2(x) - 1] - 3)^2)
Step 18: Simplify the expression within the brackets:
sin(8x) = 8sin(x)cos(x)(2sin(x)cos(x)(4cos^2(x) - 5)^2)
Step 19: Simplify further:
sin(8x) = 8sin(x)cos(x)(8sin(x)cos(x)(16cos^4(x) - 40cos^2(x) + 25))
Step 20: Factor out common terms:
sin(8x) = 8sin(x)cos(x)(8sin(x)cos(x)(16cos^4(x) - 40cos^2(x) + 25))
Finally, we have the expression sin(8x) = 8sin(x)cos(x)cos(2x)cos(4x) fully expanded and simplified.
To prove the equation sin(8x) = 8sin(x)cos(x)cos(2x)cos(4x), we can apply the double-angle and triple-angle identities to simplify both sides of the equation.
Starting with the right-hand side (RHS), we can use the double-angle identity for sine and cosine:
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos^2(θ) - sin^2(θ)
Now we substitute the double-angle identities into the RHS:
8sin(x)cos(x)cos(2x)cos(4x)
= 8sin(x)cos(x)[cos^2(2x) - sin^2(2x)]
Next, we use the double-angle identities for sine and cosine again:
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos^2(θ) - sin^2(θ)
Applying these identities:
8sin(x)cos(x)[cos^2(2x) - sin^2(2x)]
= 8sin(x)cos(x)[cos^2(2x) - (1 - cos^2(2x))]
= 8sin(x)cos(x)[2cos^2(2x) - 1]
Now, let's work on simplifying the left-hand side (LHS) of the equation. Here, we'll use the multiple-angle identity for sine:
sin(8θ) = 8sin(θ)cos^7(θ) - 28sin^3(θ)cos^5(θ) + 56sin^5(θ)cos^3(θ) - 70sin^7(θ)
Now we substitute θ with x:
sin(8x) = 8sin(x)cos^7(x) - 28sin^3(x)cos^5(x) + 56sin^5(x)cos^3(x) - 70sin^7(x)
Comparing the LHS and RHS of the equation:
sin(8x) = 8sin(x)cos^7(x) - 28sin^3(x)cos^5(x) + 56sin^5(x)cos^3(x) - 70sin^7(x)
To prove the equation, we need to show that both sides of the equation are equal. We can do this by simplifying the expression for sin(8x) using trigonometric identities, such as the double-angle identity for sine.
However, upon simplification, the equation sin(8x) = 8sin(x)cos(x)cos(2x)cos(4x) does not hold. The original equation is incorrect.