Verify Trig Identity: Find x so 1-sin(x) = 1

So be careful with that.In summary, the conversation is about testing whether an expression is an identity using a graphing calculator. The expression is found to not be an identity and the task is to find a value for x where both sides of the expression are defined but not equal. The expression is simplified to -1 and it is determined that for all defined values of x, the expression is not equal to 1. Therefore, any defined value of x can be used to fulfill the task.
  • #1
Stevo6754
30
0

Homework Statement


Use a graphing calculator to test whether the following is an identity. If it is an identity, verify it. If it is not an identity, find a value of x for which both sides are defined but not equal.

[tex]\frac{cos(-x)}{sin(x)cot(-x)}[/tex]=1


Homework Equations


None


The Attempt at a Solution


Ok, plug in the left side for y1, right side for y2, obviously not an identity. The second part where it ask for a x value is where I am having trouble. I thought maybe simplify the left hand side and find a value for whatever that is that equals 1..

Cos(-x)/sin(X)(1/-tan(X))
Cos(x)/sin(x)(-cosx/sinx)
Cos(x)/-cos(x)
Cos(x)/1-sin(x)
Cos(x)-Sin(x)cos(x)
Factor out to get
1-sin(x)
ok so now I have 1-sin(x)=1
-sin(x)=0
so find the value where -sin(x)=0??

The teachers answer is - Not an identity, x=[tex]\pi/4[/tex]

Im clueless about the second part I guess..
 
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  • #2
so, you simplified it to the form of Cos(x)/-cos(x). This just equals -1. (your next step was incorrect, though... how did you get from -cos(x) to 1-sin(x)?)
So it obviously is not equal to one. The problem statement asks for "a value of x for which both sides are defined but not equal". We have just shown that whenever the expression is defined, the equation doesn't hold. So you just need to find a value where the expression is defined.
 
  • #3
grief said:
so, you simplified it to the form of Cos(x)/-cos(x). This just equals -1. (your next step was incorrect, though... how did you get from -cos(x) to 1-sin(x)?)
So it obviously is not equal to one. The problem statement asks for "a value of x for which both sides are defined but not equal". We have just shown that whenever the expression is defined, the equation doesn't hold. So you just need to find a value where the expression is defined.

Yeah your right should be -1, I'm not quite sure how I would go about finding a value where the expression is defined.

Anyone? That was my original question, I am not sure how to find that value.
 
Last edited:
  • #4
It's defined almost everywhere. Since cot(-x)=-cos(x)/sin(x), you need to make sure sin(x)=/=0.

Then you also need to make sure that the simplified expression, cos(x)/-cos(x) is defined, i.e. cos(x)=/=0.

That's all. There are plenty of angles for which this holds. pi/4 is one, but you can also have pi/3, 2*pi/3, etc..
 
  • #5
Stevo6754, since you're able to simplify the expression and make it [tex]\frac{cos(-x)}{sinxcot(-x)}=-1[/tex], this is telling you that for all defined values of x (they are undefined where the denominator=0) it is not equal to 1. So like grief has said, the values you're looking for are not just [itex]\pi/4[/itex] but any other value that is defined.

Oh and while [itex]cos^2x=1-sin^2x[/itex] this does not mean [itex]cosx=1-sinx[/itex] since, if you square both sides you'll get [itex]cos^2x=(1-sinx)^2=1-2sinx+sin^2x\neq 1-sin^2x[/itex]
 

Related to Verify Trig Identity: Find x so 1-sin(x) = 1

1. What is a trig identity?

A trig identity is an equation that relates different trigonometric functions to each other. These identities are used to simplify expressions and solve equations involving trigonometric functions.

2. How do you verify a trig identity?

To verify a trig identity, you need to manipulate the given equation using basic trigonometric identities and algebraic rules until it matches the expression on the other side of the equation. If both sides of the equation are equal after simplifying, then the identity is verified.

3. What is the purpose of finding x in 1-sin(x) = 1?

The purpose of finding x in this equation is to determine the value of x that satisfies the equation and thus verifies the trig identity. It also allows us to solve for unknown angles in trigonometric equations.

4. How do you find x in 1-sin(x) = 1?

To find x in this equation, we first need to isolate the trigonometric function by using algebraic rules. In this case, we can add sin(x) to both sides to get 1 = 1+sin(x). Then, we can use the inverse sine function (arcsin) to find the value of x that satisfies the equation.

5. Are there any specific values for x that satisfy the equation 1-sin(x) = 1?

Yes, there are. In this equation, any value of x that satisfies the identity of sin(x) = 0 will also satisfy the equation. This includes values such as 0, π, 2π, etc. However, there are also infinitely many other values of x that satisfy the equation, such as 0.5π, 1.5π, etc.

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