Prove that it is an identity: cos (4x)=8cos^4(x)-8cos^2(x)+1
Using the property : cos 2A = 2cos^2 A -1
cos(4x)
= 2cos^2 (2x) - 1 = 2(cos(2x))(cos(2x)) - 1 , using the same property again
= 2(2cos^2 x - 1)^2 - 1
= 2(4cos^4 x - 4cos^2 x + 1) - 1
= 8cos^4 x - 8cos^2 x + 1
= RS
To prove that the equation cos(4x) = 8cos^4(x) - 8cos^2(x) + 1 is an identity, we need to show that it holds true for all values of x.
To do this, we can start by using the double-angle formula for cosine:
cos(2θ) = 2cos^2(θ) - 1
Now, let's substitute θ with 2x:
cos(4x) = 2cos^2(2x) - 1
Next, we'll expand cos^2(2x) using the double-angle formula again:
cos^2(2x) = (cos(2x))^2
= (2cos^2(x) - 1)^2
= 4cos^4(x) - 4cos^2(x) + 1
Now, substitute this back into the original equation:
cos(4x) = 2cos^2(2x) - 1
= 2(4cos^4(x) - 4cos^2(x) + 1) - 1
= 8cos^4(x) - 8cos^2(x) + 2 - 1
= 8cos^4(x) - 8cos^2(x) + 1
As you can see, the two sides of the equation match, which proves that cos(4x) = 8cos^4(x) - 8cos^2(x) + 1 is an identity.
To recap the steps:
1. Use the double-angle formula to express cos(4x) in terms of cos(2x).
2. Apply the double-angle formula again to expand cos^2(2x).
3. Substitute the expanded formula back into the original equation.
4. Simplify to show that both sides of the equation are equal.
To prove that the equation cos(4x) = 8cos^4(x) - 8cos^2(x) + 1 is an identity, we need to show that it holds true for all values of x.
First, let's use the double angle formula for cosine to express cos(4x) in terms of cos(x):
cos(4x) = 2cos^2(2x) - 1.
Next, we'll apply the double angle formula again to expand cos^2(2x):
cos^2(2x) = (cos^2(x))^2 - (sin^2(x))^2.
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite this as:
cos^2(2x) = (cos^2(x))^2 - (1 - cos^2(x))^2.
Expanding and simplifying, we have:
cos^2(2x) = cos^4(x) - (1 - 2cos^2(x) + cos^4(x)).
cos^2(2x) = 2cos^4(x) - 2cos^2(x) + 1.
Finally, substituting this expression into the equation cos(4x) = 2cos^2(2x) - 1, we get:
2cos^4(x) - 2cos^2(x) + 1 = 8cos^4(x) - 8cos^2(x) + 1.
Simplifying further, we have:
8cos^4(x) - 8cos^2(x) + 1 = 8cos^4(x) - 8cos^2(x) + 1.
Since both sides of the equation are equal, we have proven that cos(4x) = 8cos^4(x) - 8cos^2(x) + 1 is an identity.