Simplifying with Pythagorean identites.

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In summary, the given expression can be simplified using the identities sin^2 x +cos^2 x=1 and 1+cot^2 x=cosec^2 x. By combining the terms with the same denominator and using algebraic manipulation, the final simplified form is 1.
  • #1
Corkery
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Homework Statement


Simplify:

2csc^2 - csc^4 + cot^4



Homework Equations





The Attempt at a Solution



2(1/sin^2) - 1/sin^4 + cos^4/sin^4

So here, i figured that if i re-write them, that maybe it will leave me more options, but I don't see anything that will cancel out. Then i saw that you can re-write sin^2 as 1 - cos^2, so...

2(1/1 - cos^2) - 1/sin^4 + cos^4/sin^4

I now saw or at least thought that the portion of 1 - cos^2 could cancel out one of te cos^2 from the cot^4

Now we have (assuming I am correct so far.)

2(1/-1) or -2

-2 - 1/sin^4 + cos^2/sin^4

Well that is where I couldn't do anymore, but that is most liekly caused by me doing it wrong.
 
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  • #2
The answer is 1, if you didn't already know. Try again but write sinx as S, and cosx as C; this makes the expression a whole lot more manageable. I suggest taking a factor of 1/S^2 out and then forming a single fraction. Hope this helps.
 
  • #3
[tex]sin^2 x +cos^2 x=1[/tex]
so that

[tex]1+cot^2 x=cosec^2 x[/tex]
just sub that identity into what you have and you will have less to write out instead of the sin and cos
 
  • #4
I think you just started off in the wrong direction. Go back to your first expression in terms of sines and cosines. All three terms have almost the same denominator, so with a little effort you can combine all three. Now you can use some algebra (factoring) to rearrange the numerator into something a little simpler.
 
  • #5
That is an option! haha
 

1. What are Pythagorean identities?

Pythagorean identities are mathematical formulas that involve the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. These identities are used to simplify and manipulate trigonometric expressions.

2. Why is it important to simplify with Pythagorean identities?

Simplifying with Pythagorean identities allows us to reduce complex trigonometric expressions into simpler forms that are easier to work with. This can be especially useful in solving equations or evaluating limits and derivatives in calculus.

3. What are the three main Pythagorean identities?

The three main Pythagorean identities are:

  • Sine squared identity: sin2(x) + cos2(x) = 1
  • Cosine squared identity: 1 + tan2(x) = sec2(x)
  • Tangent squared identity: 1 + cot2(x) = csc2(x)

4. How do you use Pythagorean identities to simplify trigonometric expressions?

To simplify with Pythagorean identities, you can use the identities mentioned in the previous question to rewrite the trigonometric expression in terms of sine and cosine. Then, you can use algebraic techniques to combine like terms and simplify the expression further.

5. Can Pythagorean identities be used in calculus?

Yes, Pythagorean identities are frequently used in calculus to simplify trigonometric expressions and make them easier to work with. They are also used in finding derivatives and integrals of trigonometric functions.

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