#1 (permalink)

Part 2 of a test I failed last year.



Was worth 7 marks and I had no clue what to do.

#2 (permalink)

Copied from googling it - I hate trig......

LHS = tan (x / 2 + pi / 4) = [tan (x /2) + tan (pi / 4) ] / [ 1 - tan(x / 2) tan(pi / 4) ]

LHS = [ tan (x /2) + 1 ] / [ 1 - tan(x / 2) ]
LHS = [ tan (x / 2) + 1 ] ^2 / (1 - tan^2 (x / 2) )
LHS = [ tan^2 (x /2) + 2tan (x / 2) + 1 ] / (1 - tan^2 (x / 2) )
LHS = [sec^2 (x / 2) + 2tan ( x / 2) ] / (1 - tan^2 (x / 2) )
LHS = [ 1 + 2 sin (x / 2) cos( x / 2) ] / cos x
LHS = (1 + sin x) / cos x

They stop here, but I can see it would be 1/cos x plus sinx/cosx which is sec x + tan x

#3 (permalink)

Ok I'm lost in a few places:

In line three the top was squared but only tan was squared in the bottom?

How does line 5 simplify to line 6?

God I hate proving things like this. I would honestly never get this in the time frame...

#4 (permalink)

Not sure about how line 5 simplifies to line 6 (there are just too many trig identities... lol I only know the basic ones), but I can see the line 3 was not squared but multiplied by the numerator/numerator:
[ tan (x / 2) + 1 ]/[ tan (x / 2) + 1 ]

so the denominator followed the rule such that (5-x)(5+x) = 25-x^2

#5 (permalink)

LHS = [sec^2 (x / 2) + 2tan ( x / 2) ] / (1 - tan^2 (x / 2) )
LHS = [ 1 + 2 sin (x / 2) cos( x / 2) ] / cos x

I think I got it...

By identity for denominator sec^2(x) + tan^2(x) = 1
sec^2(x/2)/(sec^2(x/2)) = 1
and 2 tan (x/2)/(sec^2(x/2)) = 2 sin (x / 2) cos( x / 2)

I can't seem to think where the bottom cos (x) is coming from though I don't have paper to think clearly.. could be something to do with half angles or something since the bottom is not cos(x/2) but cos (x)

#6 (permalink)

Hm. I guess I'll just work through this some more tomorrow in the library when I can think a little better. Thanks so far, and I'll get back to you tomorrow and see if I get it any better.

#7 (permalink)

Ugh. I still can't quite get this proof.

Like where the bottom cos x came from.


Lines 5 onwards of the proof just confused the hell out of me. Even if you add in more steps can you explain what's going on, where x/2 went to etc.

I am still really lost in this proof.