Question 624957: Verify the trig identity: sin^4(x)+cos^4 (x)=3/4 + 1/4 cos(4x).
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
As we can see, the arguments to the functions on the lefts are x's and the argument to the function on the right is 4x. So somehow we will need to change the arguments from x to 4x. To change arguments we can use a variety of Trig. formulas/properties/identitites, including: sin(A+B), sin()A-B), cos(A+B), cos(A-B), sin(2x), cos(2x) sin((1/2)x) and cos((1/2)x). (Since we only have sin and cos I' omitted the properties for changing arguments of tan. And there are other properties for sin and cos that are not always taught that I have omitted.)
One way, maybe the most natural way, is to use the sin((1/2)x) and cos((1/2)x) formulas because,as you can see:
we can see that the rights side arguments are twice the size of the left have arguments. And we are wanting to double x to 2x (and then 2x to 4x). (Note: The 0's are not usually in these formulas. There are there only because algebra.com's formula drawing software requires a number in front of the "plus or minus" sign.) Sometimes we tend to avoid using these formulas because the of complication/confusion caused by the "plus or minus". But with this problem we do not need to be concerned with this issue. Our equation is raising sin's and cos's to even powers. So "plus or minus" doesn't matter because anything raised to an even power is going to end up positive.
But how do we use sin((1/2)x) and cos((1/2)x) when we have sin(x) and cos(x) instead? This is the trickiest part of using these formulas. You become a "power user" of these formulas once you learn that the x's, A's and B's in them can be replaced by any valid mathematical expression and the formula is still true!! So we can use sin((1/2)x) and cos((1/2)x) and any sin or cos. And it will double the argument each time!
etc.
Using sin((1/2)x) and cos((1/2)x) on our sin(x) and cos(x) we get:
Simplifying:
Using cos((1/2)x) on cos(2x):
And we're finished!
FWIW, here's another solution to this problem...
Our expression has high powers of sin and cos. Among the arguments changing properties only cos(2x) has higher powers of sin or cos. There are three variations of this formula:As you can see, all three variations involve sin and/or cos squared. So we will start by rewriting the left side in terms of squared sin and cos:
Next we will take the second variation of cos(2x) and solve it for  :
And similarly solve the third variation for  . (We are doing this so can substitute in for and )
Substituting these into our expression we get:
Next we'll simplify. Squaring the two expressions on the left side (using FOIL or the  pattern. (I prefer using the pattern.)):
Adding the like terms:
We're getting closer with the arguments. We've gone from x to 2x. Since we have a cos squared, we can re-use:
with the x's replaced with 2x's to substitute in for our cos squared.
Simplifying:
And we're finished!
|
|
|
