how to solve this trigonometric identity

Question

csc^2(x)/tan(x)+cot(x)

Answer 1

    csc²(x)

---------------

tan(x)+cot(x)

 

Identities:

 

csc(x) = 1/sin(x)

tan(x) = sin(x)/cos(x)

cot(x) = cos(x)/sin(x)

 

Let's plug these in so we're dealing only with sines and cosines:

 

               1/sin²(x)

-------------------------------------

sin(x)/cos(x) + cos(x)/sin(x)

 

Put the terms in the denominator over a common denominator and add them:

 

               1/sin²(x)

-------------------------------------

[sin²(x)+cos²(x)]/sin(x)cos(x)

 

Now sin²(x) + cos²(x) = 1

 

    1/sin²(x)             1         sin(x)cos(x)      cos(x)

----------------  =  --------- * ----------------  = --------   =  cot(x)

1/sin(x)cos(x)     sin²(x)           1                sin(x)