B Given an equation, find the value of this expression

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TL;DR Summary
İf ##-1+\dfrac{2}{1-x}=\sqrt 3## then find ##\dfrac{2x}{1-x^2}\cdot \sqrt 3##
İf ##-1+\dfrac{2}{1-x}=\sqrt 3## then find ##\dfrac{2x}{1-x^2}\cdot \sqrt 3##

I can solve it with two ways:
1) directly by finding ##x##
2) Let ##x=\tan\alpha## and go on and answer is ##1.##

But I need another way. Are there an another way?
 
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littlemathquark said:
I can solve it with two ways:
1) directly by finding ##x##
2) Let ##x=\tan\alpha## and go on and answer is ##1.##

But I need another way. Are there an another way?
How are you getting an "answer" of 1? This can't be the value of x as it would make the equation undefined. The value I get for x doesn't give a value of 1 for the expression ##\frac{2x}{1 - x^2}\sqrt 3##.
 
Mark44 said:
How are you getting an "answer" of 1? This can't be the value of x as it would make the equation undefined. The value I get for x doesn't give a value of 1 for the expression ##\frac{2x}{1 - x^2}\sqrt 3##.
I mean ##\frac{2x}{1 - x^2}\sqrt 3=1##
 
littlemathquark said:
TL;DR Summary: İf ##-1+\dfrac{2}{1-x}=\sqrt 3## then find ##\dfrac{2x}{1-x^2}\cdot \sqrt 3##

But I need another way.
a) Why?
b) Explain the techniques that qualify for "another".
 
fresh_42 said:
a) Why?
b) Explain the techniques that qualify for "another".
a) Just curious
b) I don't know the techniques that qualify as "another"
 
littlemathquark said:
a) Just curious
b) I don't know the techniques that qualify as "another"
So let me make this clear: you have a precise definition of the value ##x## and want to find a value ##g(x)## without using ##x## and no idea what is allowed to be used instead?
 
fresh_42 said:
So let me make this clear: you have a precise definition of the value ##x## and want to find a value ##g(x)## without using ##x## and no idea what is allowed to be used instead?
I want to find value of g(x) using/manupulating given equation( as I said I found it using trigonometry but I wonder other ways) . I assume that not known precise value of x.
 
littlemathquark said:
I can solve it with two ways:
1) directly by finding ##x##
2) Let ##x=\tan\alpha## and go on and answer is ##1.##
I get a solution for x that checks out. If I substitute this value into ##\frac{2x}{1 - x^2}\sqrt 3##, I don't get a result of 1 for that expression.

What did you get for x?
If you replace x in the expression above by ##\tan(\alpha)##, the expression becomes ##\frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}\sqrt 3 = \sqrt 3 \tan(2\alpha)##. How does that simplify to 1?
 
Mark44 said:
I get a solution for x that checks out. If I substitute this value into ##\frac{2x}{1 - x^2}\sqrt 3##, I don't get a result of 1 for that expression.

What did you get for x?
If you replace x in the expression above by ##\tan(\alpha)##, the expression becomes ##\frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}\sqrt 3 = \sqrt 3 \tan(2\alpha)##. How does that simplify to 1?
I found ##x=2-\sqrt 3## and evaluated it in ##(2x/(1-x^2)).\sqrt 3##
 
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  • #10
littlemathquark said:
I found ##x=2-\sqrt 3## and evaluated it ##(2x/(1-x^2)).\sqrt 3##
##(x+1)/(1-x)=\sqrt 3## and Let ##x=\tan\alpha## so ##(1+\tan\alpha)/(1-\tan\alpha)=\sqrt 3## and ##1=\tan 45##
##\tan(45+\alpha)=\sqrt 3=\tan 60##
##\alpha=15^{\circ} ##
 
  • #11
littlemathquark said:
I found ##x=2-\sqrt 3## and evaluated it in ##(2x/(1-x^2)).\sqrt 3##
Rechecking my work, I get the same value for x as you did.
 
  • #12
\frac{2x}{1-x^2}=\frac{1}{1- x}-\frac{1}{1+x}with\frac{1+x}{1-x}=\sqrt{3}seems useful.
 
  • #13
anuttarasammyak said:
\frac{2x}{1-x^2}=\frac{1}{1- x}-\frac{1}{1+x}with\frac{1+x}{1-x}=\sqrt{3}seems useful.
I can't use your hint. Can you give me more detail?
 
  • #14
\frac{2x}{1-x^2}=\frac{1}{1- x}-\frac{1}{1+x}=(1-\frac{1}{\sqrt{3}})\frac{1}{1-x}=(1-\frac{1}{\sqrt{3}})\frac{\sqrt{3}+1}{2}=\frac{1}{\sqrt{3}}
 
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  • #15
Thank you. Now, I have three ways for solution the problem.
 
  • #16
Geometry of
2-\sqrt{3}=tan \frac{\pi}{12}
HI-20250315_11173985.jpg
 
  • #17
anuttarasammyak said:
Geometry of
2-\sqrt{3}=tan \frac{\pi}{12}
View attachment 358523
littlemathquark said:
Thank you. Now, I have three ways for solution the problem.
Fourth solution:
##\dfrac{1+x}{1-x}=\sqrt3## so ##(1+x,1-x)=(\sqrt3k,k)## and ##k=\sqrt3-1##

##2x=(1+x)-(1-x)## and ##1-x^2=(1+x)(1-x)## so ##\dfrac{(\sqrt3-1)k}{\sqrt3k\cdot k}\cdot \sqrt 3=1##
 
  • #18
\frac{2\sqrt{3}x}{1-x^2}:=A
Ax^2+2\sqrt{3}x-A=0
x=\frac{-\sqrt{3}\pm\sqrt{A^2+3}}{A}=2-\sqrt{3}
We observe
A=1 with + sign satisfies the relation.
 
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