Sophia D.

asked • 04/03/16

Prove the following identities

1.  (sec x-1)(sec x+1) = tan2x
 
 
 
 
2.   cot 2θ =  cot2  θ-1
                   -------------             (theta)
                     2 cot θ 
 
 
3.    sin α               1
   ------------ =  ---------- (symbol is alpha)
  sin α - cos α      1- cot α

John G.

Is there supposed to be a + in one of those (secx -1) in the first one?  Don't think it'll work the way it's written.
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04/03/16

Sophia D.

No, that's the exact problem.
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04/03/16

Michael J.

Sophia, the first problem is not written correct.  As John mentioned, there should be a + in the (secx + 1) term.  Let me show you by working out the first one for you.  You will see, that the equation cannot be proven.  Let me show you what I mean.
 
 
Work out the left side.
 
(sec x - 1)(sec x - 1) =
 
sec2x - 2sec x + 1 
 
 
Using the fact that              sec x = 1 / cos x ,
 
(1 / cos2x) - (2 / cos x) + 1 =
 
(1 - 2cos x + cos2x) / cos2x
 
 
Up to this point, you know the numerator part needs to be sin2x.  This is because sin2x divided by cos2x will give you tan2x, which is the right side of your identity.  However, the numerator term is not equal to sin2x. 
 
Therefore, you have copied down the identity incorrectly.  When copying down problems on the board, take your time copy them.  Make sure you written them down correctly before leaving the classroom.
 
 
The correct identity should be
 
(sec x - 1)(sec x + 1) = tan2x
 
 
When you work out the left side, you will see that that left side matches the right side.
 
 
On that note, the others might have some errors too.
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04/03/16

Sophia D.

I just realized my mistake thank you for pointing it out the problem is (secx-1)(secx+1)=tan2x
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04/03/16

1 Expert Answer

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John G. answered • 04/03/16

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K, so if it ends up being a +, the first one's pretty simple.  You can just multiply out the parentheses and then use
sec2x -1 = tan2x   (which come from sin2x + cos2x = 1, just dividing everything by cos2x, then moving things around)
 
for the second one, I would use that tan(2x) = 2tanx/(1-tan2x), which means that cot(2x) = (1-tan2x)/2tanx
Multiply everything in the fraction by cot2x, and you should be able to get it to equal what you need to.
 
third one, leave the left side alone, replace cotx with cosx/sinx.  get the denominator to be a single fraction by finding a common denominator and then 1/that fraction is the same as the reciprocal of the fraction, which should equal the left side.
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