Prove that cos3x/cosx-cos6x/cos2x=2(cos2x-cos4x)
cos3x = cosx(2cos2x - 1)
similarly,
cos6x = cos2x(2cos4x - 1)
I think you can see where this is going, right?
Yes
To prove the given equation:
cos(3x)/cos(x) - cos(6x)/cos(2x) = 2(cos(2x) - cos(4x))
We can simplify the left-hand side of the equation using trigonometric identities, and then compare it with the right-hand side. Let's start by rewriting the left-hand side:
cos(3x)/cos(x) - cos(6x)/cos(2x)
To simplify the fractions, we need to find a common denominator. The common denominator will be cos(x) * cos(2x). Now, let's rewrite the equation:
[(cos(3x) * cos(2x)) - (cos(x) * cos(6x))] / (cos(x) * cos(2x))
Now, let's expand the numerator:
[(cos(x + 2x) * cos(x - 2x)) - (cos(x) * cos(2x))] / (cos(x) * cos(2x))
Using the cosine addition and subtraction identities, we can simplify further:
[(cos(3x) * (cos(x) * cos(2x) - sin(x) * sin(2x))) - (cos(x) * cos(2x))] / (cos(x) * cos(2x))
Now, let's factor out the common term of (cos(x) * cos(2x)):
cos(x) * cos(2x) * [cos(3x) - 1] / (cos(x) * cos(2x))
Notice that the term (cos(x) * cos(2x)) cancels out. We are left with:
cos(3x) - 1
Now, let's compare it with the right-hand side of the equation:
2(cos(2x) - cos(4x))
Using cosine subtraction identity, we can simplify:
2(-2sin(x)sin(3x))
Now, we can compare the simplified left-hand side (cos(3x) - 1) with the right-hand side (2(-2sin(x)sin(3x))). We can see that they are not equal.
Therefore, the given equation cos(3x)/cos(x) - cos(6x)/cos(2x) = 2(cos(2x) - cos(4x)) is not valid.
In this case, we have proven that the equation is not true by simplifying each side separately and comparing the results.