Pre Calc Help?

Question

Solve the proofs:

1) csc^4x - cot^4x = 2csc^2 - 1

2) cos^4x - cos2x
____________ = -sinx
2sin3x
**Note** For the second problem, that ____ line is supposed to be a divided by sign
any help is appreciated

Answer 1

1) Strategy: convert everything to csc². Key identity: csc²x - 1 = cot²x.
csc^4x - cot^4x
(csc²x - cot²x)(csc²x + cot²x)
(csc²x - (csc²x - 1))(csc²x + (csc²x - 1))
2csc²x - 1

2) Strategy: convert everything to sin.
cos^4x - cos2x
-------------------
2sin3x

cos²x cos²x - (1 - 2sin²x)
-------------------------------
2sin(x+2x)

(1-sin²x)(1-sin²x) - 1 + 2sin²x
-------------------------------------
2(sinx cos2x +cosx sin2x)

sin^4x - 2sin²x + 1 - 1 + 2sin²x
-------------------------------------------------
2 sinx (1 - 2sin²x) + cosx (2 sinx cosx)

sin^4x
------------------------------------
2sinx - 2sin³x + 2sinx cos²x

sin^4x
-------------------------------------
2sinx - 2sin³x + 2sinx(1 - sin²x)

sin^4x
-------------------------------------
2sinx - 2sin³x + 2sinx - 2sin³x

sin^4x
-----------------
4sinx - 4sin³x

sin^4x
-------------------
4sinx(1 - sin²x)

sin³x
--------
4cos²x

gah, now i'm stuck, and out of time. :(

Answer 2

since (x+y)(x-y0=x^2-y^2 so:left hand side:
=(csc^2x+cot^2x)(csc^2x-cot^2x... .........1)
now ID: cot^2x = csc^2x - 1
or csc^2x - cot^2x = 1
=>1)....becomes:
(csc^2x+cot^2x) = csc^2x + cos^2xcsc^2x
=csc^2x(1+cos^2x)
=csc^2x(1+1-sin^2x)
=csc^2x(2-sin^2x)
=2csc^2x-1
proved
2. cos4x - cos2x = -2sin3xsinx

ID: cos(A+B)-cos(A-B)= -2sinAsinB
so RHS: A=3x and B=x
=> -2sin3xsinx = cos4x -cos 2x
proved.

Answer 3

csc^4x-cot^4x = 2csc^2 x -1
1/sin^4x -(cos^4x)(sin^4x) = 2/sin^2x -1
=(1-cos^4x)/sin^4x =(2-sin^2x)/sin^2x
=(1-cos^4x)/sin^4x =(1+1-sin^2x)/sin^2x
=(1-cos^2x)(1+cos^2x)/sin^4x = (1+cos^2x)(sin^2x/sin^4
So (1-cos^2x)(1+cos^2x) = (sin^2x)(1+cos^2x)
1-cos^2x =sin^2x
1= cos^2x + sin^2x
1=1

(cos^4x - cos2x)/(2sin3x) = - sinx
Sorry, Wife is taking computer.

Answer 4

1.(csc^2x+cot^2x)(csc^2x-cot^2x)
=csc^2x+cot^2x(1)
=csc^2x+(csc^2x-1)
=2csc^2x-1
hence proved

the second problem is not clear

Answer 5

hmm yeah good luck with that one...