Verifying (Pre Calculus)?

Question

I have one problem. I understand how to "verifying," but I am completely puzzled with this one. I have spent atleast 15 minutes on this problem and have not gotten anywhere.

1) 1-cos/1+cos = (cot - csc)(cot - csc)

Again, this is a verifying problem.
Please, I would really appreciate some help here...thanks!!

Answer 1

1-cos/1+cos

1-cosx (1-cosx)
--------- * ------------
1+cosx (1-cosx)

1-2cosx+cos^2x
-----------------------
1-cos^2x

1-2cosx+cos^2x
------------------------
sin^2x

1/sin^2x - 2cosx/sinx + cos^2x/sin^2x
csc^2x -2secxcotx + cot^2x
factor that and you get
(cot-csc)(cot-csc)

Answer 2

I started off by multiplying the left side by [(1-cos)/(1-cos)]. This will give you [(1-2cos+cos^2)/(1-cos^2)]
= [(1-2cos+cos^2)/sin^2]
=1/sin^2 - 2cos/sin^2 + cos^2/sin^2
= csc^2 - 2cotcsc + cot^2
=(csc - cot)(csc - cot)
= (-1)(cot - csc) (-1) cot - csc)
= (cot - csc)(cot - csc)

Answer 3

1-cos/1+cos = 1-cos/1+cos
<==>
1-cos/1+cos = (1-cos/1+cos)*(1-cos/1-cos)
<==>
1-cos/1+cos = (1-2cos+cos^2) / (1-cos^2)
<==>
1-cos/1+cos = (1-2cos+cos^2) / sin^2
<==>
1-cos/1+cos = cos^2/sin^2 – 2cos/sin^2 + 1/sin^2
<==>
1-cos/1+cos = cot^2 – 2csc*cot + csc^2
<==>
1-cos/1+cos = (cot – csc)(cot - csc)

Answer 4

the 2d step is faulty. in case you had, say, a / (b+c), that does no longer mean that that is the same as a(a million/b + a million/c). you won't be able to invert such issues as that once you've a "+" in there. you should address b+c as a unmarried time period, whose reciprocal isn't inevitably (a million/b)+(a million/c). as an social gathering, a million/4 = a million/(2+2), yet a million/2 + a million/2 = a million, no longer 4. attempt utilizing the very undeniable truth that cot(x) = cos(x)/sin(x), and csc(x) = a million/sin(x) (a million+csc(x)) / (cot(x)+cos(x)) (a million + (a million/sin(x))) / ([cos(x)/sin(x)] + cos(x)) sin(x)(a million + (a million/sin(x))) / sin(x)([cos(x)/sin(x)] + cos(x)) (sin(x) + a million) / (cos(x) + sin(x)cos(x)) (sin(x) + a million) / cos(x)(a million+sin(x)) a million / cos(x) sec(x)

Answer 5

(1-cosx)/(1+cosx) = (cot - csc)(cot - csc)
(1-cosx)(1-cosx)/[(1-cosx)(1+cosx)]=cot^2x -2cotxcscx+csc^x)
(1-2cosx+cos^2x)/(1-cos^2x) =cot^2x -2cotxcscx+csc^x)
(1-2cosx+cos^2x)/(sin^2x) =cot^2x -2cotxcscx+csc^x)
csc^2x -2cotxcscx cot^2x = cot^2x -2cotxcscx+csc^x)