SOLUTION: Trigonometry Identities Prove the identity is true 1/cosx - cosx = sinx tanx

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Question 95110: Trigonometry Identities
Prove the identity is true

1/cosx - cosx = sinx tanx
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Here's one way. You will need to use the two identities:
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tan%28x%29+=+sin%28x%29%2Fcos%28x%29
and
cos%5E2%28x%29+%2B+sin%5E2%28x%29+=+1
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Given:
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1%2F%28cos%28x%29%29+-+cos%28x%29+=+sin%28x%29%2Atan%28x%29
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substitute sin%28x%29%2Fcos%28x%29 for tan(x) to get:
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1%2F%28cos%28x%29%29+-+cos%28x%29+=+sin%28x%29%2A%28sin%28x%29%2Fcos%28x%29%29
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This substitution changes the right side and the equation becomes:
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1%2F%28cos%28x%29%29+-+cos%28x%29+=+sin%5E2%28x%29%2Fcos%28x%29
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Get rid of the denominator cos%28x%29 by multiplying all the terms on both sides to get:
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cos%28x%29%2F%28cos%28x%29%29+-+cos%5E2%28x%29+=+cos%28x%29%2Asin%5E2%28x%29%2Fcos%28x%29
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Cancel the cos%28x%29 in the denominator with the corresponding cos%28x%29 in the numerator:
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and this reduces the equation to:
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1+-+cos%5E2%28x%29+=+sin%5E2%28x%29
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Add cos%5E2%28x%29 to both sides to get rid of the -cos%5E2%28x%29 on the left side. This
changes the equation to:
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1+=+cos%5E2%28x%29+%2B+sin%5E2%28x%29
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This is in the form of a trigonometric identity that you are probably familiar with already.
Since you have now worked the original given equation into a form known to be true, you
have successfully proved that the original given equation is true.
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Hope that this helps you to understand the problem.
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