verifying trig identity: (CosX+1)/(tan^2x) = (CosX)/(secX-1)
- Thread starter ChrisRR
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Is there any significance to the capitalization...?
To clarify, do you mean the following?
. . . . .\(\displaystyle \L \frac{\cos{(x)}\, +\, 1}{\tan^2{(x)}}\,=\, \frac{\cos{(x)}}{\sec{(x)}\, -\, 1}\)
If so, then try simplifying each side separately. You should be able to get each side down to "cos<sup>2</sup>(x) / [1 - cos(x)]". You can then do the proof by working down one side to this point, and then working backwards up the other side to get to that side's starting point.
Eliz.
To clarify, do you mean the following?
. . . . .\(\displaystyle \L \frac{\cos{(x)}\, +\, 1}{\tan^2{(x)}}\,=\, \frac{\cos{(x)}}{\sec{(x)}\, -\, 1}\)
If so, then try simplifying each side separately. You should be able to get each side down to "cos<sup>2</sup>(x) / [1 - cos(x)]". You can then do the proof by working down one side to this point, and then working backwards up the other side to get to that side's starting point.
Eliz.
Re: verifying trig identity: (CosX+1)/(tan^2x) = (CosX)/(sec
IF you recognize (from the fundamental identities) that sec^2 x - 1 = tan^2 x, then working with the right-hand side should be obvious.
Multply numerator and denominator of the right-hand side by (sec x + 1):
cos x(sec x + 1)
---------------
(sec x - 1)(sec x + 1)
cos x(sec x + 1)
-----------------
sec^2 x - 1
cos x(sec x + 1)
----------------
xxxtan^2 x
OK....now we have the denominator we want. Distribute the multiplication in the numerator:
cos x sec x + cos x
----------------
xxxtan^2 x
1 + cos x
----------
xtan^2 x
There you go! That is what is on the right-hand side, so you're done!
I hope this helps you.
ChrisRR said:
IF you recognize (from the fundamental identities) that sec^2 x - 1 = tan^2 x, then working with the right-hand side should be obvious.
Multply numerator and denominator of the right-hand side by (sec x + 1):
cos x(sec x + 1)
---------------
(sec x - 1)(sec x + 1)
cos x(sec x + 1)
-----------------
sec^2 x - 1
cos x(sec x + 1)
----------------
xxxtan^2 x
OK....now we have the denominator we want. Distribute the multiplication in the numerator:
cos x sec x + cos x
----------------
xxxtan^2 x
1 + cos x
----------
xtan^2 x
There you go! That is what is on the right-hand side, so you're done!
I hope this helps you.