Question on proving an identity

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In summary, the conversation discusses a problem where the goal is to prove the identity 2csc(2x)=csc^2(x)tanx. The speaker shares their attempt at solving the problem, but got stuck and looked at the answer in the back of their pre-calculus book. The book's solution shows that 2csc(2x) can be simplified to 1/sin^2(x) multiplied by sinx/cosx, which equals csc^2(x)tanx. The speaker then asks for clarification on how 2csc(2x) equals 1/sin^2(x) multiplied by sinx/cosx.
  • #1
Lazz72
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So I'm given a problem in which I have to prove an identity. It goes:
2csc2x=csc^2xtanx

I did the problem myself and could only get to 2csc2x=2\(sin2x)= 2\(2sinxcosx). I had no idea how to get further with the problem so I looked at the answer in the back of my pre-calculus book. It said that:

2\(2sinxcosx) = 1\(sin^2x) multiplied by sinx\cosx = csc^2xtanx

I don't understand the part that says 2\(2sinxcosx)= 1\(sin^2x) multiplied by sinx/cosx. Could someone please explain to me how 2\(2sinxcosx) equals that?
 
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  • #2
Hi Lazz72, welcome to PF.
2csc(2x) = 2/sin(2x) = 2/2six*cosx = 1/sinx*cos* = sinx/sinx*sinx*cosx = tanx/sin^2(x)
 
  • #3
Ahh okay I understand it now, thanks!
 

1. What is the process for proving an identity?

The process for proving an identity involves starting with one side of the equation and manipulating it using algebraic properties and rules until it is equivalent to the other side of the equation. This shows that both sides of the equation are equal and the identity is proven.

2. How do I know which algebraic properties to use when proving an identity?

When proving an identity, it is important to remember the basic algebraic properties such as the commutative, associative, and distributive properties. In addition, trigonometric identities may require knowledge of special angle relationships and identities.

3. Can I use any values for the variables in an identity?

No, the values for the variables should be chosen carefully to ensure that the identity holds true for all values of the variables. This is especially important when working with trigonometric identities, as certain values may result in undefined expressions.

4. What should I do if I get stuck while trying to prove an identity?

If you get stuck while trying to prove an identity, it can be helpful to review the basic algebraic properties and rules, as well as any relevant trigonometric identities. It may also be helpful to simplify both sides of the equation separately before trying to manipulate them.

5. Are there any tips for making the process of proving an identity easier?

One helpful tip for proving an identity is to work backwards from the desired outcome. Start with the side of the equation that you want to prove, and manipulate it using algebraic properties and rules until it is equivalent to the other side of the equation. Another tip is to practice regularly, as the more familiar you become with different identities and their proofs, the easier the process will become.

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