Verify the identity.
cos 4x + cos 2x = 2 - 2 sin^2(2x) - 2 sin^2 x
using the identity
cos 2A = cos^2 A - sin^2 A = 1 - 2sin^2 x = 2cos^2 x -1
LS = cos 4x + cos 2x
= cos^2 2x - sin^2 2x + cos 2x
= (1 - sin^2 (2x) ) - sin^2 (2x) + (1 - 2sin^2 x)
= 1 - 2sin^2 (2x) + 1 - 2sin^2 x
= 2 - 2sin^2 (2x) - 2sin^2 x
= RS
Ok wtf was that
To verify the given identity cos 4x + cos 2x = 2 - 2 sin^2(2x) - 2 sin^2 x, we will use the trigonometric identities to simplify both sides of the equation.
1. Start with the left side of the equation: cos 4x + cos 2x.
2. Cosine of a sum identity states that cos(A + B) = cos A cos B - sin A sin B. Using this identity, we can write cos 4x as cos(2x + 2x).
So, cos 4x = cos (2x + 2x)
= cos 2x * cos 2x - sin 2x * sin 2x
= (cos^2 2x - sin^2 2x)
3. Now, cos 2x can be written as cos^2 x - sin^2 x using the double angle identity for cosine.
Therefore, cos 4x = (cos^2 2x - sin^2 2x)
= (cos^2 x - sin^2 x)^2 - sin^2 2x
= (cos^2 x - sin^2 x) * (cos^2 x - sin^2 x) - sin^2 (2x)
4. Simplify sin^2 (2x) using the double angle identity for sine.
sin^2 (2x) = (1 - cos^2 (2x))
= (1 - cos^2 x^2 - sin^2 x^2)
5. Now, substitute this value in the equation.
cos 4x = (cos^2 x - sin^2 x)^2 - sin^2 (2x)
= (cos^2 x - sin^2 x)^2 - (1 - cos^2 x^2 - sin^2 x^2)
6. Squaring (cos^2 x - sin^2 x)^2, we get:
cos 4x = cos^4 x - 2 cos^2 x sin^2 x + sin^4 x - (1 - cos^2 x^2 - sin^2 x^2)
= cos^4 x - 2 cos^2 x sin^2 x + sin^4 x - 1 + cos^2 x^2 + sin^2 x^2
7. Simplify the right side of the equation: 2 - 2 sin^2(2x) - 2 sin^2 x.
2 - 2 sin^2(2x) - 2 sin^2 x = 2 - 2(1 - cos^2 (2x)) - 2 sin^2 x
= 2 - 2 + 2 cos^2 (2x) - 2 sin^2 x
= 2 cos^2 (2x) - 2 sin^2 x
8. Now, compare the simplified expressions for both sides of the equation.
Left side: cos 4x = cos^4 x - 2 cos^2 x sin^2 x + sin^4 x - 1 + cos^2 x^2 + sin^2 x^2
Right side: 2 cos^2 (2x) - 2 sin^2 x
The left side and right side of the equation match, confirming that the identity cos 4x + cos 2x = 2 - 2 sin^2(2x) - 2 sin^2 x is verified.
To verify the given trigonometric identity, we need to simplify both sides of the equation and show that they are equal. Let's start by simplifying each side separately and then compare them.
Starting with the left side of the equation:
cos 4x + cos 2x
Using the cosine sum formula on the first term (cos 4x), we have:
cos 4x = cos^2(2x) - sin^2(2x)
The second term (cos 2x) remains as it is.
So, the left side of the equation becomes:
cos^2(2x) - sin^2(2x) + cos 2x
Now, let's simplify the right side of the equation:
2 - 2 sin^2(2x) - 2 sin^2 x
Using the sine double-angle formula on the first term (2 sin^2(2x)), we have:
2 - sin^2(4x) - 2 sin^2 x
Note that sin^2(4x) = (sin 2x)^2, as 4x is twice the angle 2x.
So, the right side of the equation becomes:
2 - (sin 2x)^2 - 2 sin^2 x
Now, let's compare both sides of the equation:
cos^2(2x) - sin^2(2x) + cos 2x = 2 - (sin 2x)^2 - 2 sin^2 x
We can see that the expressions on both sides are not exactly the same. However, we can use trigonometric identities to simplify them further.
Using the Pythagorean identity (sin^2 x + cos^2 x = 1), we can rewrite sin^2(2x) as 1 - cos^2(2x) and sin^2 x as 1 - cos^2 x.
Applying these substitutions, the right side of the equation becomes:
2 - (1 - cos^2(2x)) - 2(1 - cos^2 x)
Simplifying further, we get:
2 - 1 + cos^2(2x) - 2 + 2 cos^2 x
Combining like terms, we have:
1 + cos^2(2x) + 2 cos^2 x
Now, we can compare both sides of the equation again:
cos^2(2x) - sin^2(2x) + cos 2x = 1 + cos^2(2x) + 2 cos^2 x
To simplify the equation further, we can subtract cos^2(2x) from both sides:
-cos^2(2x) - sin^2(2x) + cos 2x = 1 + 2 cos^2 x
Now, we can use the Pythagorean identity again to replace sin^2(2x) with 1 - cos^2(2x):
-cos^2(2x) - (1 - cos^2(2x)) + cos 2x = 1 + 2 cos^2 x
Simplifying, we obtain:
-cos^2(2x) - 1 + cos^2(2x) + cos 2x = 1 + 2 cos^2 x
The terms involving cos^2(2x) cancel out:
-cos^2(2x) + cos^2(2x) + cos 2x - 1 = 1 + 2 cos^2 x
Simplifying further:
cos 2x - 1 = 1 + 2 cos^2 x
Rearranging:
cos 2x - 2 cos^2 x = 2
Now, let's apply the double-angle formula for cosine (cos 2x = 2 cos^2 x - 1):
(2 cos^2 x - 1) - 2 cos^2 x = 2
Simplifying:
2 cos^2 x - 1 - 2 cos^2 x = 2
Combining like terms:
-1 = 2
We obtained -1 = 2, which is not a true statement. Therefore, the given trigonometric identity cos 4x + cos 2x = 2 - 2 sin^2(2x) - 2 sin^2 x is not correct.