SOLUTION: Simplify do not solve (cos x)/(csc x - sin x)

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Question 515391: Simplify do not solve (cos x)/(csc x - sin x)
Answer by drcole(72) About Me  (Show Source):
You can put this solution on YOUR website!
Recall that csc+x+=+%281%29%2F%28sin+x%29. So we substitute this into our expression and work from there:
+%28cos+x%29%2F%28csc+x+-+sin+x%29+
+%28cos+x%29%2F%28%281%29%2F%28sin+x%29+-+sin+x%29%29 (substituting)
+%28cos+x%29%2F%28%281%29%2F%28sin+x%29+-+%28sin+x%29%5E2%2F%28sin+x%29%29 (putting all of the terms in the denominator under a common denominator)
+%28cos+x%29%2F%28%281+-+%28sin+x%29%5E2%29%2F%28sin+x%29%29 (combining fractions in the denominator)
+%28cos+x%29+%2A+%28%28sin+x%29%2F%281+-+%28sin+x%29%5E2%29%29 (simplifying the complex fraction by multiplying by the reciprocal of the denominator)
Now recall that %28cos+x%29%5E2+%2B+%28sin+x%29%5E2+=+1, so %28cos+x%29%5E2+=+1+-+%28sin+x%29%5E2. So, substituting %28cos+x%29%5E2 for 1+-+%28sin+x%29%5E2, we get:
+%28cos+x%29+%2A+%28%28sin+x%29%2F%28%28cos+x%29%5E2%29%29 (substituting)
+1+%2A+%28%28sin+x%29%2F%28cos+x%29%29 (canceling one cos+x)
+tan+x+ (remembering that tan+x+=+%28sin+x%29%2F%28cos+x%29)
So our expression simplified to tan+x.