SOLUTION: Please help. How can I prove the identity? Is there any online calculator for trig identity? Thank you. 2cscx=tan(1/2)x+cot(1/2)x How to prove the identity?

Algebra ->  Trigonometry-basics -> SOLUTION: Please help. How can I prove the identity? Is there any online calculator for trig identity? Thank you. 2cscx=tan(1/2)x+cot(1/2)x How to prove the identity?      Log On


   



Question 1107091: Please help.
How can I prove the identity? Is there any online calculator for trig identity?
Thank you.
2cscx=tan(1/2)x+cot(1/2)x
How to prove the identity?

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
There is no online calculator that will solve identities
like this:

2csc%28x%29=tan%28expr%281%2F2%29x%29%2Bcot%28expr%281%2F2%29x%29

Let's get the right tools:

There are two formulas for tan%28expr%281%2F2%29theta%29

which don't involve square roots.  They are:

tan%28expr%281%2F2%29theta%29%22%22=%22%22%281-cos%28theta%29%29%2Fsin%28theta%29

and

tan%28expr%281%2F2%29theta%29%22%22=%22%22sin%28theta%29%2F%281%2Bcos%28theta%29%29

and since the cotangent is the reciprocal of the tangent, we can
make two formulas from them for cot%28expr%281%2F2%29theta%29

cot%28expr%281%2F2%29theta%29%22%22=%22%22sin%28theta%29%2F%281-cos%28theta%29%29

and  

cot%28expr%281%2F2%29theta%29%22%22=%22%22%281%2Bcos%28theta%29%29%2Fsin%28theta%29

Let's pick two of them for the right side of the identity which
have the same denominator so they'll be easy to add.  Pick the 
ones with sin(x) denominators to substitute in the right side 
of the identity, which is:

tan%28expr%281%2F2%29x%29%2Bcot%28expr%281%2F2%29x%29

Substituting:

%281-cos%28x%29%29%2Fsin%28x%29%22%22%2B%22%22%281%2Bcos%28x%29%29%2Fsin%28x%29

%28%281-cos%28x%29%5E%22%22%29%2B%281%2Bcos%28x%29%5E%22%22%29%29%2Fsin%28x%29%5E%22%22

%281-cos%28x%29%5E%22%22%2B1%2Bcos%28x%29%5E%22%22%29%2Fsin%28x%29%5E%22%22



2%2Fsin%28x%29

2%2A%281%2Fsin%28x%29%29

2csc%28x%29

Edwin