Use the fudamental identities to write the expression in terms of a single trigonometric function.?

Question

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

Answer 1

2 Csc(x)

To keep this simple, let
Sec(x) = 1/c
Csc(x) = 1/s
Cot(x) = c/s
We have:
(1/c) / (1/s - c/s) - (1/c) / (1/s + c/s)
((1/c)(1/s + c/s) - (1/c)(1/s - c/s)) / (1/s² - c²/s²)
((1/c)(2c/s)) / (1/s²)(1 - c²)
(2/s) / (1/s²)(s²)
2/s = 2 Csc(x)

Answer 2

I assume the denominator of the first fraction is csc x - cot x, and the denominator of the second fraction is csc x + cot x, even though as written, it technically isn't.

Multiply the top and bottom of the first fraction by csc x + cot x. this is en route to a common denominator (the product of the two denominators). The top of the first fraction becomes

sec x (csc x + cot x),

which can simplify to sec x csc x + csc x. the denominator becomes

(csc x - cot x )(csc x + cot x),

which FOILs to give csc^2 x - cot^2 x, which equals 1 (use a Pythagorean identity. Nice, the denominator is 1. So the first fractions boils down to

sec x csc x + csc x

Now, do the same with the second fraction, except multiply the top and bottom of it by (csc x - cot x). Again, the denominator of the second fraction becomes 1, but the top becomes

sec x csc x - csc x

Because the second fraction is subtracted, watch your signs.

The whole problem is now

(sec x csc x + csc x) - (sec x csc x - csc x)

Clear out parentheses, and you should be down to one trig function.


ta da.

Answer 3

I interpreted what you wrote as such I omitted the x's to make typing a little easier. I will put the x back in at the end.

[sec/(csc-cot)] - [sec/(csc+cot)]

Treat it lit two fractions. Find the common denominator and write a one fraction

sec(csc+cot) - sec(csc-cot)
-------------------------------------
(csc-cot)(csc+cot)


Multiply out the tops and bottoms of fractions

sec(csc) + sec(cot) - sec(csc) + sec(cot)
------------------------------------------------------------
csc^2 - cot^2

Simply the top first

2 sec(cot)
-----------------
csc^2 - cot^2

Use the fundamental identity cot^2 + 1 = csc^2 and get it to look the bottom of the fraction you get [1 = csc^2 - cot^2]; Replace the bottom with 1 you end up with

2 sec(cot)

Using the fundamental identies sec = (1/cos) and cot = (cos/sin)

2 * cos
----------------
cos * sin

eliminate what they have in common you get

2
---
sin x

you can rewrite this as

2cscx

Answer 4

(sec(x) / csc(x) - cot(x)) - (sec(x) / csc(x) + cot(x))
= sec(x) / csc(x) - cot(x)) - sec(x) / c sc(x) - cot(x) = -2 cot(x)
Or U mean :
(sec(x) / (csc(x) - cot(x)) - (sec(x) /( csc(x) + cot(x))
= sec(x) [ 1/(csc(x) - cot(x)) - 1/ ( csc(x) + cot(x))]
= sec(x) [(csc(x) + cot(x) - csc(x) + cot(x))/( csc^2(x) - cot^2(x)]
= sec(x) * [ 2 cot(x)/1]
= 2 /cos(x) * cos(x)/sin(x)
= 2/sin(x)=2 csc(x)