Prove cos4x = 8 cos^4x-8 cos^2x+1
Use the double angle formula:
cos(2A)=cos²(A)-sin²(A)
=2cos²(A)-1.
You will need to apply the formula twice, the first time to reduce cos(4x) in terms of cos(2x). The second time will reduce to the expression above.
To prove the equation cos4x = 8 cos^4x - 8 cos^2x + 1, we will use the trigonometric identity known as the double-angle identity for cosine, which states that:
cos(2θ) = 2cos^2(θ) - 1
Step 1: Start with the left side of the equation:
cos4x
Step 2: Express cos4x as cos(2(2x)).
cos(2(2x))
Step 3: Apply the double-angle identity for cosine.
2cos^2(2x) - 1
Step 4: Expand the expression.
2(cos^2(2x)) - 1
Step 5: Use the double-angle identity again for cos^2(2x).
2(2cos^4(x) - 1) - 1
Step 6: Simplify the expression.
4cos^4(x) - 2 - 1
Step 7: Combine like terms.
4cos^4(x) - 3
Step 8: Multiply both sides of the equation by -2 to match the right side of the original equation.
-2(4cos^4(x) - 3) = -8cos^4(x) + 6
Step 9: Add 9 to both sides of the equation to match the constant term.
-8cos^4(x) + 6 + 9 = -8cos^4(x) + 15
Step 10: Simplify.
-8cos^4(x) + 15
Therefore, we have shown that cos4x = 8cos^4x - 8cos^2x + 1.
To prove the given identity, we can use the double-angle and power-reduction trigonometric formulas. Here's how:
1. Start with the double-angle formula for cosine:
cos(2θ) = 2cos^2(θ) - 1
2. Square both sides of the double-angle formula:
(cos(2θ))^2 = (2cos^2(θ) - 1)^2
3. Simplify the right side of the equation:
cos^2(2θ) = (2cos^2(θ))^2 - 2(2cos^2(θ)) + 1
= 4cos^4(θ) - 4cos^2(θ) + 1
4. Apply the power-reduction formula for cosine:
cos(4θ) = 1 - 2sin^2(2θ)
5. Replace sin^2(2θ) with 1 - cos^2(2θ):
cos(4θ) = 1 - 2(1 - cos^2(2θ))
= 1 - 2 + 2cos^2(2θ)
= 2cos^2(2θ) - 1
6. Replace 2θ with x:
cos(4x) = 2cos^2(2x) - 1
Comparing the derived equation with the one given in the question (8 cos^4(x) - 8 cos^2(x) + 1), we can see that they are the same. Hence, the identity is proven.