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April 21st, 2018, 06:39 AM   #1
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Question Determine values of variables for 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

Last edited by greg1313; April 21st, 2018 at 01:36 PM.
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April 21st, 2018, 07:10 AM   #2
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2.
$$1-\frac{\sin(2x)}{\sin(x)} - \cos(x)=\sin(x)-\cos(x)$$
if and only if
$$1-\frac{2\sin(x)\cos(x)}{\sin(x)} = \sin(x)$$
if and only if
$$1-2\cos(x) = \sin(x)$$
Letting $$t=\tan(x/2)$$, we get
$$1-2\frac{1-t^2}{1+t^2} = \frac{2t}{1+t^2}$$
if and only if
$$1+t^2 - 2+2t^2 = 2t$$
if and only if
$$3t^2 - 2t-1=0$$
Discriminant is $$D=7$$
So
$$t = \frac{2\pm \sqrt{7}}{6}$$
So
$$x = 2\text{arctan}\left(\frac{2\pm \sqrt{7}}{6}\right)$$
Now I'm sure you can do the rest in the same way!

Last edited by skipjack; April 26th, 2018 at 08:28 AM.
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April 21st, 2018, 10:00 AM   #3
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The second equation lacked parentheses; it should have been $\frac{1 - \sin 2x}{\sin x - \cos x} = \sin x - \cos x$.
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April 21st, 2018, 01:38 PM   #4
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Quote:
Originally Posted by Vee88 View Post
1. 2x / Cosx- Sinx = Cosx + Sinx

2.1 - Sin 2x / Sinx- Cosx = Sinx - Cosx

3.Sinx + Sinx /1 + Cosx + Cos2x = Tanx
Hi Vee88 and welcome to the forum.

Can you clarify your work by adding parentheses where appropriate?
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