Trigonometric proof using derivatives

In summary, the task is to prove that for every x smaller than -1, the equation \frac{1}{2}arctan\frac{2x}{1-x^{2}}+arccotx=\pi holds. The solution involves splitting the equation into two parts, calculating their derivatives separately, and showing that they are both equal to \frac{1}{x^{2}+1}. However, it is important to note that just because the derivatives are equal, it does not necessarily mean the original equations are equal. It is necessary to consider the domain of the functions and add additional steps to prove the equation holds for all values in that domain. Additionally, the constant C must be calculated to fully prove the equation
  • #1
Pole
9
0

Homework Statement


Prove that (for every x smaller than -1)
[itex]\displaystyle \frac{1}{2}arctan\frac{2x}{1-x^{2}}+arccotx=\pi[/itex]


Homework Equations





The Attempt at a Solution


So i split the formula into two parts:
[itex]\displaystyle \frac{1}{2}arctan\frac{2x}{1-x^{2}}[/itex] and [itex]\displaystyle \pi-arccotx[/itex]
Calculated their derivatives separately
And they are both equal to [itex]\displaystyle \frac{1}{x^{2}+1}[/itex]

And I'm wondering whether it's the end of the task or should I prove some other things, especially something connected to the main assumption of x<-1 or any other.

Thanks in advance!
 
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  • #2
If [itex]f^\prime=g^\prime[/itex], then this doesn't mean f=g necessarily. But it does mean that f=g+C with C a real constant/

Note that the above only holds if the domain of f and g is an interval.
 
  • #3
micromass said:
If [itex]f^\prime=g^\prime[/itex], then this doesn't mean f=g necessarily. But it does mean that f=g+C with C a real constant/

Note that the above only holds if the domain of f and g is an interval.

Thank You, now I know that it isn't sufficient.
But I can't really come up with a complete solution.
What should I add ? So that the task would be marked well
How to calculate the value of that constant?
 

FAQ: Trigonometric proof using derivatives

What is a trigonometric proof using derivatives?

A trigonometric proof using derivatives is a mathematical method of proving identities involving trigonometric functions by using the rules of differentiation.

Why is using derivatives helpful in trigonometric proofs?

Using derivatives allows us to manipulate and simplify complex trigonometric expressions, making it easier to prove identities.

How do you use derivatives to prove trigonometric identities?

To prove a trigonometric identity using derivatives, we start by taking the derivative of both sides of the equation and then manipulate the resulting expressions until they are equal.

What are some common derivatives used in trigonometric proofs?

Some common derivatives used in trigonometric proofs include the derivatives of sine, cosine, and tangent, as well as the chain rule and product rule.

Are there any limitations to using derivatives in trigonometric proofs?

Yes, derivatives can only be used to prove identities that involve continuous functions. They cannot be used to prove identities involving discrete values, such as the values of trigonometric ratios for specific angles.

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