Expert: Ahmed Salami - 2/9/2009

Question
So I'm having trouble solving this problem.

1. Prove the Identity
         Sin2x= 1/(tanx+cot2x)

Could you run me through the steps of solving. I keep getting stuck. It would really help to know where I went wrong. Thanks


Answer
Hi Brooke,
Remember the double-angle identities
sin2x = 2sinx.cosx
cos2x = cos²x - sin²x
and
sin²x + cos²x = 1

cot2x = 1/tan2x
     = cos2x/sin2x
Now,
tanx + cot2x = sinx/cosx + cos2x/sin2x
            = sinx/cosx + [(cos²x - sin²x)/(2sinx.cosx)]
            = [2sinx.sinx + (cos²x - sin²x)]/(2sinx.cosx)
            = [2sin²x + (cos²x - sin²x)]/(2sinx.cosx)
            = [sin²x + cos²x]/(2sinx.cosx)
but,
sin²x + cos²x  = 1
2sinx.cosx = sin2x
so,
tanx + cot2x = 1/sin2x
Therefore,
sin2x = 1/(tanx + cot2x)

Regards