proving identities?

Question

we are on proving identities in trigonometry and i dont understand this

sec + csc
---------- == sec
1 + cot

how do i get one side equal to the other, can you explain to me???

Answer 1

Pull out a common sec out of the numerator:

sec + csc
-------------
1 + cot

sec(1 + csc/sec)
-----------------------
1 + cot

Now notice that csc = 1/sin and sec = 1/cos

So csc/sec = 1/sin / 1/cos

This is the same as cos/sin which is also the same as cot

Finally you get:
sec( 1 + cot )
-------------------
(1 + cot)

Now you can cancel the 1 + cot to get:

sec

QED

Answer 2

sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
cot(x) = cos(x)/sin(x) = csc(x)/sec(x)

LHS
= (sec(x) + csc(x)) / (1 + cot(x))
= (sec(x) + csc(x)) / (1 + csc(x)/sec(x))
= sec(x)(sec(x) + csc(x)) / (sec(x) + csc(x))
= sec(x)
= RHS