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sin(x+a)=sqrt(2)cos(x-a): find tan x
By Anonymous on Wednesday, May 31, 2000 - 01:35 am:

I haven't got a clue how to prove the following question. Can anyone help?

Given that sin(x+a) =sqrt(2)cos(x-a), where cosx×cosa is not equal to 0,

a] prove that tanx = sqrt(2)-tan(a)/(1-sqrt(2)tan(a))

Thanks

By Harry Smith (Harry) on Wednesday, May 31, 2000 - 11:53 am:

You will need to use your compound angle formulae to do this question. If you see something like cos(x+a) it's always a good idea to see if you can use any identities you know already. Using 2 instead of root 2 (it makes no difference), your original equation was:

sin (x+a)=2cos (x-a)

Expanding this out using compound angle formulae gives you:

sin x cos a + cos x sin a = 2cos x cos a + 2sin x sin a

Now we can try and find a clue about what to do next from the original question. The question states that cos x cos a is not equal to 0. This suggests that you will need to divide by cos x cos a at some point. Do this to both sides of the equation, remembering that tan x = sin x/cos x, and you have:

tan x + tan a = 2 + 2tan x tan a.

You are now essentially done. Collect the terms with tan x on one side, factor our tan x, and divide both sides by the other factor.
You now have:

tan x = (2 - tan a)/(1 - 2tan a)

as required.

Hope this is clear.
Harry

By Anonymous on Wednesday, May 31, 2000 - 12:16 pm:

Harry,

That was very clear, thanks very much.

I was able to expand the compound angle formula, but didn't know what to do next. I can see the usefulness of hints such as the product not being equal to 0. I'll remember that. It also made common sense to use 2 instead of the squareroot of 2! I'll remember that too.

Thanks again Harry