My Math Forum Determine the values of the variables for which each of the following identities
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 April 23rd, 2018, 03:13 AM #1 Newbie   Joined: Apr 2018 From: East London Posts: 12 Thanks: 0 Determine the values of the variables for which each of the following identities 1.{2x / Cosx- Sinx} = Cosx + Sinx 2.{1 - Sin 2x}/ {Sinx- Cosx} = Sinx - Cosx 3.{Sinx + Sinx} /{1 + Cosx + Cos2x} = Tanx Determine the values of the variables for which each of the following identities are undefined...... Please help i just need get through matric
 April 23rd, 2018, 05:02 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 I'm afraid that doesn't quite make sense. First, an "identity" is an equation that is true for all values of x and so must be defined for all values of x. What you have are not identities but equations. But the word "equation" doesn't make sense here either because it is an "expression", perhaps the left of right side of an equation, that is or is not defined. Finally, I don't know what "matric" means. I thought perhaps "matrix" or "metric" but neither of those has anything to do with this problem. Sine and Cosine themselves are defined for all x. Tan(x) is undefined for x equal to odd multiples of $\displaystyle \frac{\pi}{2}$. Fractions are undefined when the denominator is undefined.
 April 23rd, 2018, 05:35 AM #3 Newbie   Joined: Apr 2018 From: East London Posts: 12 Thanks: 0 Ok I'll see my lecturer today because that's all he gave to us ..Thank you
April 23rd, 2018, 05:43 AM   #4
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Quote:
 Originally Posted by Vee88 1.{2x / Cosx- Sinx} = Cosx + Sinx 2.{1 - Sin 2x}/ {Sinx- Cosx} = Sinx - Cosx 3.{Sinx + Sinx} /{1 + Cosx + Cos2x} = Tanx Determine the values of the variables for which each of the following identities are undefined...... Please help i just need get through matric
1. can only be solved numerically.

2. is actually an identity. (Multiply both sides by sin(x) - cos(x).)

I have not found a simple way to handle 3.

-Dan

April 23rd, 2018, 06:56 AM   #5
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Quote:
 Originally Posted by topsquark 2. is actually an identity. (Multiply both sides by sin(x) - cos(x).)
It is not though. It doesn't hold for $x=\pi/4$.

April 23rd, 2018, 07:07 AM   #6
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Quote:
 Originally Posted by Micrm@ss It is not though. It doesn't hold for $x=\pi/4$.
Well, okay, except for values that aren't allowed in the original expression.

Thanks for the catch.

-Dan

April 25th, 2018, 12:59 PM   #7
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Quote:
 Originally Posted by Vee88 3.{Sinx + Sinx} /{1 + Cosx + Cos2x} = Tanx
$\displaystyle \dfrac{2 \sin x}{1+\cos x +(2\cos^2x-1)}=\tan x$

$\displaystyle \dfrac{2\sin x}{\cos x(2\cos x+1)}=\tan x$

$\displaystyle 2\tan x = \tan x(2\cos x+1)$

$\displaystyle \tan x(2\cos x+1)-2\tan x = 0$

$\displaystyle \tan x(2\cos x-1)=0$. Now you can finish it.

 April 26th, 2018, 01:09 AM #8 Global Moderator   Joined: Dec 2006 Posts: 20,101 Thanks: 1905 Was the first equation intended to be $\displaystyle \frac{\cos 2x}{\cos x - \sin x} = \cos x + \sin x$? That holds except when $x = \pi/4 + \text{k}\pi$, where k is an integer.

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