[(sec x) / (csc x - cot x)] - [(sec x) / (csc x + cot x)] = 2 csc x?

Question

verify the identity. make the left side look like the right side.

Answer 1

Ok. Let's simplify the left side of the equation.

1. For the first term, [(sec x) / (csc x - cot x)], you multiply (csc x + cot x). For the second term, [(sec x) / (csc x + cot x)], you multiply (csc x - cot x).

2. Alright, now you simplify it. You should get secXcscX+secXcotX-secXcscX+secXcotX= 2 cscX.

3. Once you get that, combine like terms. You should now have [(2 secXcotX)/(cscX^2 - cotX^2) = 2 cscX.

4. Convert all the trig signs on the left to another form(like, cscX = (1/sinX) ). On the top of the left side, you should get 2 * (1/cosX)(cosX/sinX).

5. You cross multiply the cos X's to cancel, which will give you 2 * (1/sinX). On the bottom you should have (1/sinX^2) - [(cosX^2)/(sinX^2)], which would simplify out to [(1-cosX^2)/(sinX^2)].

6. Now that you have the top and bottom of the fraction on the left side, remember what you do when you're diving fractions? You multiply by the reciprocal of the 2nd term. So, that's what you do. You should write down 2 (1/sinX) * [(sinX^2)/(1-cosX^2)] = 2 cscX.

7. Ignore the 2 for now, and just simplify the fractions. You see how the sinX's cross multiply to give you 2 (sinX)/(1-cosX^2) = 2 cscX?

8. Ok. Now remember the identity, "sinX^2 + cosX^2 = 1"? The bottom of our fraction on the left is just that identity phrased a different way. Simply subtract cosX^2 from both sides and you get sinX^2 = (1 - cosX^2), right?

9. Go back to our equation on the left side. Well what do you know? We have (1-cosX^2) right there. That equals sinX^2, so now you should write down 2 (sinX)/(sinX^2).

10. Simplify the sinX's, and you get 2 (1)/(sinX) = 2 cscX. Voila, your answer.

Here's a step by step break down:

1. [secX(cscX+cotX) - secX(cscX-cotX)]/[(cscX^2+cscXcotX-cscXcotX-cotX^2)] = 2 cscX

-> [(secXcscX+secXcotX - secXcscX-secXcotX)/(cscX^2-cotX^2)] = 2 cscX

2. [(secXcotX+secXcotX)/(cscX^2-cotX^2)] = 2 cscX.

-> [(2 secXcotX)/(cscX^2-cotX^2)] = 2 cscX

3. Just the top part of the fraction:
[2 (1/cosX)(cosX/sinX)]
-> Cross multiply the cosX to cancel out, get:
[2 (1/sinX)]

Now the bottom part of the fraction:
(1/sinX^2) - (cosX^2)/(sinX^2)
-> since denominator is same, put them together into:
[(1 - cosX^2)/(sinX^2)]

4. Dividing fractions = multiplying by 2nd term's reciprocal, so:
[2 (1/sinX)] / [(1 - cosX^2)/(sinX^2)] =
{[2 (1/sinX)] * [(sinX^2)/(1-cosX^2)]}

-> Put aside the 2 for now, and only simplify the fractions. You see you can only cross multiply the SinX?
[2 (sinX)/(1-cosX^2)] = 2 cscX

5. Remember the identity sinX^2 + cosX^2 = 1?
That can be changed to 1-cosX^2 = sinX^2.
Seem familiar? Right. We have 1-cosX^2 in our fraction. Simply plug in the SinX^2 to the 1-cosX^2:
[2 (sinX)/(sinX^2)] = 2 cscX

6. You see how you can simplify that fraction further?
[2 (1)/(sinX)] = 2 cscX

-> Remember cscX = (1/sinX)?

2 cscX = 2 csc X
There's your answer.

Answer 2

I don't have the time, but if you combine the fraction on the left over (csc^2 x - cot^2 x), your numerator will beome something like 2sec x cot x. The sec x cot x can be converted to a denominator of cos x tan x which can make something interesting happen.......

Answer 3

element out sec sec(x)(-csc^2(x) + cot(x)tan(x)) = -csc(x)cot(x) sec(x)(-csc^2(x) + a million ) = -csc(x)cot(x) sec(x)(-csc^2(x) + a million ) = -csc(x)cot(x) sec(x)( -cot^2(x) ) = -csc(x)cot(x) sec(x) ( -cos^2/sin^2 ) = -csc(x)cot(x) a million/cos ( -cos^2/sin^2 ) = -csc(x)cot(x) -cos/sin^2 = -csc(x)cot(x) -csc(x)cot(x) = -csc(x)cot(x) i'm hoping this facilitates