SOLUTION: Add or subtract the fractions then simplify using the Pythagorean identities and factoring to find the expression that completes the identity. sinx/(cosx+1) + (cosx+1)/sin x

Algebra ->  Algebra  -> Trigonometry-basics -> SOLUTION: Add or subtract the fractions then simplify using the Pythagorean identities and factoring to find the expression that completes the identity. sinx/(cosx+1) + (cosx+1)/sin x       Log On

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Question 140791: Add or subtract the fractions then simplify using the Pythagorean identities and factoring to find the expression that completes the identity.
sinx/(cosx+1) + (cosx+1)/sin x
Please show me the steps. Thank you!
Answer by solver91311(12126) About Me  (Show Source):
You can put this solution on YOUR website!
I can show you how to add the fractions, but since there is no equal sign anywhere, there is no "identity"

You add the fractions just like you would add any other fractions. You find a common denominator, convert the numerators, and then sum the numerators.

%28sin%28x%29%2F%28cos%28x%29%2B1%29%29%28%28cos%28x%29%2B1%29%2Fsin%28x%29%29

The two denominators have no factors in common, so the lowest common denominator is simply the product of the denominators, exactly like adding 1%2F5%2B1%2F7 where the LCD is just 5%2A7=35

So your LCD is sin%28x%29%28cos%28x%29%2B1%29 and your sum becomes:

%28sin%5E2%28x%29%2B%28cos%28x%29%2B1%29%5E2%29%2F%28sin%28x%29%28cos%28x%29%2B1%29%29

%28sin%5E2%28x%29+%2B+cos%5E2%28x%29%2B2cos%28x%29%2B1%29%2F%28sin%28x%29%28cos%28x%29%2B1%29%29

But we know that sin%5E2%28x%29%2Bcos%5E2%28x%29=1 so:

%281%2B2cos%28x%29%2B1%29%2F%28sin%28x%29%28cos%28x%29%2B1%29%29

%282cos%28x%29%2B2%29%2F%28sin%28x%29%28cos%28x%29%2B1%29%29

cartoon%28%282%28cos%28x%29%2B1%29%29%2F%28sin%28x%29%28cos%28x%29%2B1%29%29%2C%282%2Across%28%28cos%28x%29%2B1%29%29%29%2F%28sin%28x%29cross%28%28cos%28x%29%2B1%29%29%29%29


2%2Fsin%28x%29

2csc%28x%29