Expert: Sherman D. - 4/10/2009

Question
please help!!!  this problem is killing me after 4 pages of different attempts...

Verify the trigonomic identity:

(cot x - tan x) / (sin x + cos x) = csc x - sec x

thank you for your help!

Answer
(cot(x) - tan(x))/(sin(x) + cos(x)) = csc(x) - sec(x)

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

Multiply everything at the top by sin(x)cos(x)

((cos(x)^2 - sin(x)^2)/(sin(x)cos(x)))/(sin(x) + cos(x))

now just think of this like this
(x/y)/z = (x/y)/(z/1) = (x/y)*(1/z) = x/(yz)

so your problem now becomes

(cos(x)^2 - sin(x)^2)/((sin(x)cos(x))(sin(x) + cos(x)))

now just factor the top and you get

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

as you can see the (cos(x) + sin(x))s cancel out, leaving you with

(cos(x) - sin(x))/(sin(x)cos(x))

to make it more understandable, you can put this as

(cos(x)/(sin(x)cos(x))) - (sin(x)/(sin(x)cos(x)))

now just take out like values and you have

(1/sin(x)) - (1/cos(x))

which is the same as

csc(x) - sec(x)

more info found at http://math2.org/math/trig/identities.htm

sometimes you have to put them certain ways as long as they mean the same thing, then treat your values as variables so that you can understand it better. But that really only works when you have like values.