Can You Simplify csc(θ) - sin(θ) to cos(θ)cot(θ)?

• lwelch70
In summary, the conversation discusses a method for proving the identity csc(theta) - sin(theta) = cos(theta)*cot(theta) by manipulating each side of the equation using known identities. The conversation also highlights the importance of finding a common denominator in order to simplify the equation.
lwelch70
csc(theta) - sin(theta) = cos(theta)*cot(theta)

I'm supposed to write a proof for this but to be honest I'm not really sure where I should even start. The prof taught to take one side of the equation and simply manipulate each part into its equivalent until the other side of the equation was reached (if that makes sense). I just can't seem to get it flowing.

Some work I'm trying,

I know that csc(theta) = 1/sin(theta). But if I substitute for this I simply get [1/sin(theta)] - sin(theta). I'm just really stuck and could use a start as to how to go about proving this.

actually this identity isn't that challenging

as you said csc = 1/sin *i'm not going to keep saying theta, you should know it's there

so then you'll have 1/sin - sin, find common denominator by just multiplying sin by sin/sin to get (1 - sin^2)/sin

1-sin^2 is just cos^2, so know you have (cos^2)/sin, which is just cos(cot)

understand?

Muliply both sides by some thing that will make it look more familiar.

physicsman2 said:
actually this identity isn't that challenging

as you said csc = 1/sin *i'm not going to keep saying theta, you should know it's there

so then you'll have 1/sin - sin, find common denominator by just multiplying sin by sin/sin to get (1 - sin^2)/sin

1-sin^2 is just cos^2, so know you have (cos^2)/sin, which is just cos(cot)

understand?

iamthegelo said:
Muliply both sides by some thing that will make it look more familiar.

Okay guys, that helps me out alot. I guess I was on the right track I'm just too out of it to realize that I needed to multiply to get a common denominator. Thanks again for the guidance

no problem all it takes is some manipulation and knowledge of other identities to get these

1. How do you prove a trigonometric identity?

There are several methods for proving trigonometric identities, including using algebraic manipulations, using the unit circle, and using trigonometric identities themselves.

2. Can all trigonometric identities be proven?

No, there are some trigonometric identities that cannot be proven. These are known as non-provable identities, and they are usually derived from the Pythagorean theorem.

3. What is the purpose of proving trigonometric identities?

Proving trigonometric identities helps to verify mathematical equations and relationships, and also allows for simplification and transformation of trigonometric expressions.

4. How can I check if my proof of a trigonometric identity is correct?

You can check your proof by substituting values for the variables in the original identity and in your proof, and seeing if they produce the same result. You can also use a graphing calculator or math software to graph both sides of the equation and see if they overlap.

5. Are there any tips for solving trigonometric identities?

Some tips for solving trigonometric identities include simplifying expressions using known identities, using substitution to change variables, and working with one side of the equation at a time. It is also helpful to have a good understanding of basic trigonometric functions and their properties.

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