Ammarah
  • Ammarah
Identities HELP FINAL TOMORROW
Mathematics
jamiebookeater
  • jamiebookeater
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Ammarah
  • Ammarah
|dw:1433373713959:dw|
Ammarah
  • Ammarah
perl
  • perl
we should start with the right side?

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Ammarah
  • Ammarah
yess
perl
  • perl
$$ \Large { \tan x ( 1 + \cos(2x) )= \\ = \tan x ( 1 + 2\cos^2(x)-1 )= } $$
perl
  • perl
i used the substitution cos(2x) = 2cos^2 x -1
Ammarah
  • Ammarah
how did u get rid of the cosx>
perl
  • perl
$$ \Large { \tan x ( 1 + \cos(2x) ) \\ = \tan x ( 1 + \color{red}{ 2\cos^2(x)-1} ) \\= \tan x \cdot 2\cos^2(x) \\= \frac{\sin x}{\cos x } \cdot 2\cos^2(x) \\~\\= 2 \sin x \cos x \\~\\= \color{blue}{\sin(2x)} } $$
perl
  • perl
the red and blue are substitutions
perl
  • perl
Trig identities: sin(2x) = 2sin x cos x cos(2x) = 2cos^2 x - 1
Ammarah
  • Ammarah
i know but u know u cosx was left
Ammarah
  • Ammarah
how did u get rid of that..
perl
  • perl
in the last line, 2sin x cos x = sin(2x) , thats a trig identity
perl
  • perl
You can prove that by use of the addition formula: sin(2x) = sin(x+x) = sin x cos x + cos x sin x = 2 sin x cos x
perl
  • perl
this is a good resource for identities http://www.purplemath.com/modules/idents.htm
Ammarah
  • Ammarah
ohkk
Ammarah
  • Ammarah
i have another queestion
Ammarah
  • Ammarah
|dw:1433375868645:dw|
perl
  • perl
Is this the question? $$ \Large \csc x = \frac{\cot ( x/2)}{1 + \cos x } $$
Ammarah
  • Ammarah
yes
perl
  • perl
Let's use the half angle identity.
perl
  • perl
tan (x/2) = sin x / ( 1 + cos x )
perl
  • perl
cot(x/2) = 1 / tan(x/2) = ( 1 + cos x ) / sin x
perl
  • perl
thats an equivalent identity
perl
  • perl
cot(x/2) / ( 1 + cos x ) =cot(x/2) * 1 / ( 1 + cos x ) = (1 + cos x ) / sin x * 1 / ( 1 + cos x ) = 1 / sin x = csc x
perl
  • perl
I used this identity http://prntscr.com/7cuud8
Ammarah
  • Ammarah
ok thanks

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