how do I simplify these?
1. (Cot/1-tan) + (Tan/1-Cot) - Tan - Cot
2. (1+cos) (csc-cot)
sloppy notation.
the cot of what, the sin of what?
sin, cos, tan, etc are mathematical operators
I will use x as the "angle"
(Cotx/1-tanx) + (Tanx/1-Cotx) - Tanx - Cotx
I usually change everybody to sines and cosines, so ...
= (cosx/sinx)/(1 - sinx/cosx) + (sinx/cosx)/(1 - cosx/sinx) - sinx/cox - cosx/sinx
= (cos^2 x)/(sinx(cosx - sinx)) + (sin^2x)/(cosx(sinx-cosx) - sinx/cosx - cosx/sinx
= (cos^2 x)/(sinx(cosx - sinx)) - (sin^2x)/(cosx(cosx - sinx) - sinx/cosx - cosx/sinx
= .lots of messy typing here
form a common denominator of
(sinx)(cosx)(cosx-sinx) and try to finish it.
for the second, I would use the same approach.
(1+cosx) (cscx-cotx)
= (1 + cosx)(1/sinx - cosx/sinx)
= 1/sinx - cosx/sinx + cosx/sinx - cos^2x/sinx
= 1/sinx(1 - cos^2x)
= 1/sinx(sin^2x_
= sinx
(cos/sin/(1 - sin/cos) + sin/cos/(1-cos/sin) - sin/cos - cos /sin
multiply top and bottom of all by sin cos
cos^2/(sin cos -sin^2) + sin^2/(sin cos - cos^2) - (sin^2+cos^2)/sin cos
cos^2/(sin cos -sin^2) + sin^2/(sin cos - cos^2) - 1 /sin cos
cos^2/(sin (cos -sin)) - sin^2/(cos(cos - sin)) - 1 /sin cos
writing cos as c and sin as s
just doing first two terms for now
c^2/(s(c-s)) - s^2/(c(c-s))
1/(c-s) * (c^2/s -s^2/c)
1/(c-s) * ( (c^3-s^3)/sc)
1/(c-s) * ((c-s)(c^2 + sc + s^2)/sc
(1+sc)/sc
now put that -1/sc back
(1+sc)/sc - 1/sc
sc/sc
1 Caramba !!!!
It is the sins, cos, etc of theta
OK, I was not about to type theta all the time either, and in fact even got tired of typing sin and cos.
To simplify the given expressions, we'll follow the order of operations and apply trigonometric identities where necessary.
1. (Cot/1-tan) + (Tan/1-Cot) - Tan - Cot
Let's simplify each term one by one:
a) Cot/1-tan:
Using the reciprocal identity, Cot(x) = 1/Tan(x), we can rewrite this as 1/Tan(x) / (1 - Tan(x)).
To simplify further, we multiply the numerator and denominator by Tan(x):
[(1/Tan(x)) * Tan(x)] / [(1 - Tan(x)) * Tan(x)] = 1 / (Tan(x)-Tan^2(x))
b) Tan/1-Cot:
Using the reciprocal identity, Tan(x) = 1/Cot(x), we can rewrite this as 1/Cot(x) / (1 - Cot(x)).
To simplify further, we multiply the numerator and denominator by Cot(x):
[(1/Cot(x)) * Cot(x)] / [(1 - Cot(x)) * Cot(x)] = 1 / (Cot(x)-Cot^2(x))
c) Tan - Cot:
There are no identities to apply here. We keep it as is.
Now, let's substitute back the simplified expressions:
[1 / (Tan(x)-Tan^2(x))] + [1 / (Cot(x)-Cot^2(x))] - Tan(x) - Cot(x)
Since Tan^2(x) + 1 = Sec^2(x) and Cot^2(x) + 1 = Csc^2(x), we can substitute these identities:
[1 / (Tan(x) - Sec^2(x))] + [1 / (Cot(x) - Csc^2(x))] - Tan(x) - Cot(x)
Now, we have simplified the expression as much as possible.
2. (1+cos) (csc-cot)
Let's distribute the terms:
(1+cos) * csc - (1+cos) * cot
To simplify each term, we'll use reciprocal identities and distribute:
a) (1+cos) * csc:
Using the reciprocal identity, csc(x) = 1/sin(x), we can rewrite this as (1+cos) * (1/sin(x)), which becomes (1+cos) / sin(x).
b) (1+cos) * cot:
Using the reciprocal identity, cot(x) = 1/tan(x), we can rewrite this as (1+cos) * (1/tan(x)), which becomes (1+cos) / tan(x).
Now, we have simplified the expression to:
(1+cos) / sin(x) - (1+cos) / tan(x)
Note: Depending on the context, further simplification might be possible, but these are the primary steps to simplify the given expressions using trigonometric identities.