Help verifying trigonometric identities?

Question

I already tried to solve this one:

cot^4 X + cot^2X = csc^4 X - csc^2 X

I couldn't get to the answer, what I did was:

cot^2 X ( cot^2 X + 1)
cot^2 X - csc^2 X

cos^2 X csc^2 X
--------- - ---------
sin^2 X

But I can't get any further, any suggestions or ideas on how to solve this ?

Answer 1

You know that 1 + cot^2 x = csc^2 x, so csc^2 x -1 = cot^2 x.

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

Answer 2

Okay, yes the identity cot^2(x) + 1 = csc^2(x) but since it's put in the parenthesis that needs to be multiplied by cot^2(x) not subtracted from it. So you get

cot^2(x)c­sc^2(x)

Next you'll want to simplify them into sin and cos

[cos^2(x) / sin^2(x)] [1 / sin^2(x)] = [cos^2(x) / sin^4(x)]

Now we can go to the right side of the equation. You can proceed as you did before, pulling out the common elements.

csc^4(x) - csc^2(x)
csc^2(x)[csc^2(x) - 1]

Using the trig identity we know that csc^2(x) - 1 = cot^2(x) and so we can substitute it.

csc^2(x)[cot^2(x)]

Again we'll simplify to sin and cos

[1 / sin^2(x)] [cos^2(x) / sin^2(x)] = [cos^2(x) / sin^4(x)]

This is what we also got for the left side.

Answer 3

I shall start at the point where you left

(cos^2x/sin^2x) csc^2 x (do not disturb csc^2 x as it is on rhs)
= cot^2x csc^2 x
= (csc^2 x -1) csc^2 x (convert to csc^2x as cot is not on RHS)
= csc^4 x - csc^2 x

Answer 4

in proving you should only manipluate one side... if possible.

try to use factoring!
cot^4 x= (cotx)^4 and so as cot^2 x= (cotx)^2
factor out cot^2 x
cot^2 x(cot^2 x+ 1)
cot^2 x ( cos^2 x over sin^2 x +1)

lcd...
cot^2( cos^2x+ sin^2x all over sin^2 x)
simplify
cot^2x over sin^2x
cos^2x over sin^4 x
but since u cnat some out with the other side manipulate it too.
factor out csc^2x(csc^2-1)= csc^2x(cot^2x)--> pythagorean identity

cos^2x over sin^4x= csc^2x(cot^2x)
cos^2xover sin^4x=cos^2xover sin^4x

Answer 5

cot^4(x) + cot^2(x) = csc^4(x) - csc^2(x)

cos^4(x) / sin^4(x) + cos^2(x) / sin^2(x) = 1/sin^4(x) - 1/sin^2(x)
multiply both sides by sin^4(x)

cos^4(x) + sin^2(x)cos^2(x) = 1 - sin^2(x)
cos^2(x) * [cos^2(x) + sin^2(x)] = cos^2(x)
cos^2(x) * 1 = cos^2(x)
cos^2(x) = cos^2(x)

Answer 6

wrong question,verify csc