anonymous
  • anonymous
rewrite terms in sine and cosine only: sec X - cosX/(1+sin X)
Mathematics
jamiebookeater
  • jamiebookeater
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Michele_Laino
  • Michele_Laino
we can substitute this: \[\sec x = \frac{1}{{\cos x}}\]
Michele_Laino
  • Michele_Laino
so we can write: \[\sec x - \frac{{\cos x}}{{1 + \sin x}} = \frac{1}{{\cos x}} - \frac{{\cos x}}{{1 + \sin x}}\]
anonymous
  • anonymous
how would i simplify it?

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Michele_Laino
  • Michele_Laino
hint: the common denominator is: \[\cos x\left( {1 + \sin x} \right)\]
Michele_Laino
  • Michele_Laino
what is: \[\frac{{\cos x\left( {1 + \sin x} \right)}}{{\cos x}} = ...?\]
anonymous
  • anonymous
how is cos the denominator?
Michele_Laino
  • Michele_Laino
it is an intermediate step
Michele_Laino
  • Michele_Laino
the denominator of our new equivalent fraction is: \[{\cos x\left( {1 + \sin x} \right)}\]
anonymous
  • anonymous
then i get \[\frac{1+\sin X-\cos ^{2}X\ }{ cosX(1+sinX) }\]
Michele_Laino
  • Michele_Laino
correct!
Michele_Laino
  • Michele_Laino
now, we can use this identity: \[{\left( {\cos x} \right)^2} = \left( {1 - \sin x} \right)\left( {1 + \sin x} \right)\]
anonymous
  • anonymous
oh yay thank you so much !!!
Michele_Laino
  • Michele_Laino
:)

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