Not sure how to approach problem

[The Bob]

(\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\times\frac{\sqrt{2}}{2})+(\frac{\sqrt{2}}{2}\times\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}})

Completely stuck here now. I can't really see where to start. Do I square verything to remove the square root signs?

The Bob (2004 ©)

 

[dextercioby]

(\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\times\frac{\sqrt{2}}{2})+(\frac{\sqrt{2}}{2}\times\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}})

Completely stuck here now. I can't really see where to start. Do I square verything to remove the square root signs?

The Bob (2004 ©)

(\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\times\frac{\sqrt{2}}{2})+(\frac{\sqrt{2}}{2}\times\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}})=(\frac{\sqrt{2-\sqrt{2}}}{2}\cdot\frac{\sqrt{2}}{2})+(\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2}}{2})
=\frac{1}{4}(\sqrt{\sqrt{8}-2}+\sqrt{\sqrt{8}+2})

Daniel.

 

[The Bob]

(\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\times\frac{\sqrt{2}}{2})+(\frac{\sqrt{2}}{2}\times\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}})=(\frac{\sqrt{2-\sqrt{2}}}{2}\cdot\frac{\sqrt{2}}{2})+(\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2}}{2})
=\frac{1}{4}(\sqrt{\sqrt{8}-2}+\sqrt{\sqrt{8}+2})

The problem is, Dex, that I cannot see your stages of getting from one set of numbers to another. If I had the equation \frac{1}{x}=\frac{1}{2} then I don't need to do any stages because it is so simple. The set of equations I have tried to simplify are simple to you but not to me. Therefore if I am to understand I need some stages in between e.g. I do not need to go:

\frac{1}{x}=\frac{1}{2}
=> 1=\frac{1}{2} x
=> \frac{1}{\frac{1}{2}}= x

=> 1 \times \frac{2}{1} = x

=> 1 \times 2 = x = 2

Can you see what I mean? Both me and you and everyone on these forums can see that the answer to x was 2 without doing the set of works I have just done.

This is the same for you with (\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\times\frac{\sqrt{2}}{2})+(\frac{\sqrt{2}}{2}\times\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}) but not for me. It my be silly to you but it isn't to me. I need those stages to help me understand.

Thanks for all the help you have given me so for.

The Bob (2004 ©)

 

[dextercioby]

Okayn,let's calculate the first term VERY EXPLICITELY and i'll let u do the second,since it's very much similar.

I_{1}=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}} \times \frac{\sqrt{2}}{2}(1)

I_{1}=A\times B (2)
,where
A=:\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}} (3)
B=:\frac{\sqrt{2}}{2} (4)

Let's compute "A":
You know that:
\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}} (5)
So let's apply it in our case
A=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}} =\frac{\sqrt{1-\frac{\sqrt{2}}{2}}}{\sqrt{2}} =\frac{\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2}}}{\sqrt{2}}=\frac{\sqrt{2-\sqrt{2}}}{2} (6)

Then the product A*B becomes
I_{1}=\frac{\sqrt{2-\sqrt{2}}}{2}\times \frac{\sqrt{2}}{2} =\frac{1}{4} C (7)
,where "C" is
C=:\sqrt{2-\sqrt{2}}\times \sqrt{2} =\sqrt{2\sqrt{2}-2}=\sqrt{\sqrt{8}-2} (8)

Therefore
I_{1}=\frac{\sqrt{\sqrt{8}-2}}{4}

Daniel.

 

[The Bob]

Okayn,let's calculate the first term VERY EXPLICITELY and i'll let u do the second,since it's very much similar.

I see how it works now, therefore, I feel it is not necessary for me to do the second part as it simply means I change a subtract sign for an addition sign.

I thank you very much.

So \sin\frac{3\pi}{8}=\frac{1}{4}(\sqrt{\sqrt{8}-2}+\sqrt{\sqrt{8}+2}). This is the answer you wanted? Oh and I can take out a common factor so, again, I did not see the need to show it.

The Bob (2004 ©)

P.S. If that is the answer you wanted then can I have another one that will mean I do not need to learn more but can simply apply what is in this thread, just to make sure I know what I am doing.

 

[dextercioby]

This is a challanging problem.I haven't done it myself,so i don't know any solution and neither the answer.
Compute via trigonometry:
\sin\frac{\pi}{5}

Daniel.

PS.I would be grateful to the one that comes up with a solution.