Power reducing formulas can be used to rewrite 8sin^2 x cos^2 x as 1 - cos4x
True?
Thanks
Preliminary:
pick any angle, (not the standard 30, 60 45 etc)
let's say x = 29°
so 8sin^2 29° x cos^2 29° = 1.43837...
1 - cos 4(29°) = 1 - cos 116° = 1.43837..
It sure looks like the statement is true.
If that is all they wanted, this would do it
But if you had to actually prove it, you got some work ahead
BTW, what I did above does not "prove" that the statement is true, it merely illustrates it.
On the other hand if the results would have been different, then it would be wrong, since all we need is one counterexample.
If on the other hand we get the same result for some unusual angle, the chances that they are equal by accident are so remote, that we can usually say with 99.99% certainty that the statement is true
thank you very much!
To determine whether the expression 8sin^2(x)cos^2(x) can be rewritten as 1 - cos(4x) using power-reducing formulas, we need to apply these formulas and compare the two expressions.
Power-reducing formulas state that sin^2(x) = (1 - cos(2x))/2 and cos^2(x) = (1 + cos(2x))/2.
Let's substitute these formulas into the given expression:
8sin^2(x)cos^2(x) = 8[(1 - cos(2x))/2][(1 + cos(2x))/2]
Now, let's simplify this expression:
8[(1 - cos(2x))/2][(1 + cos(2x))/2]
= (8/4)[(1 - cos(2x))(1 + cos(2x)])
= 2(1 - cos^2(2x))
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite cos^2(2x) as 1 - sin^2(2x):
2(1 - cos^2(2x)) = 2(1 - (1-sin^2(2x)))
= 2(1 - 1 + sin^2(2x))
= 2sin^2(2x)
Thus, the expression 8sin^2(x)cos^2(x) simplifies to 2sin^2(2x), not 1 - cos(4x). Therefore, the statement "8sin^2(x)cos^2(x) can be rewritten as 1 - cos(4x) using power-reducing formulas" is FALSE.