I need help with a trig proof. I've tried for a long time and can't get it. cot^2x=csc^2x-sin^2x-cos^2x

Question

It seems like I've tried everything, but apparently I haven't

Answer 1

well its easy
cot^2x=csc^2x-sin^2x-cos^2x
cot^2x=csc^2x-(sin^2x+cos^2x)
cot^2x=csc^2x-1
triangle proof
csc^2x = cot^2x + 1
check pythagorean identities

Answer 2

On the right, factor out a -1:

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

cos^2 x / sin^2 x = csc^2 x - 1(1)

cos^2 x / sin^2 x = csc^2 x - 1

cos^2 x / sin^2 x = 1/ sin^2 x - sin^2 x/sin^2 x

cos^2 x / sin^2 x = (1 - sin^2x) / sin^2 x

cos^2 x / sin^2 x = cos^2 x / sin^2 x

Answer 3

First, note that -sin^2x -cos^2x = -1
so you're really trying to prove that cot^2x = csc^2x - 1

cscx = 1/sinx
and cotx = cosx/sinx

So, cot^2x = cos^2/sin^2 = 1/sin^2x - 1 is equivalent to the previous expression.

multply both sides by sin^2 x

cos^2x = 1 - sin^2x
Add sin^2x to both sides, and we have
cos^2x + sin^2x = 1
which is the pythagorean theorem!

Answer 4

ok so you wanna keep one side the same and then make the other side equal that. So in this question, its easier to keep the left side the same.

So you have csc^2(x) - sin^2(x) - cos^2(x).
You kno the identity:
csc^2(x) = 1/sin^2(x).

Now you have (1/sin^2(x)) - sin^2(x) - cos^2(x). So you put everything over sin^2(x), which would give you:

(1-sin^4(x) - cos^2(x)sin^2(x))/sin^2(x)
Now factor by -sin^2(x) to give you:
(1-sin^2(x) * (sin^2(x) +cos^2(x)))/sin^2(x)

We know that sin^2(x) + cos^2(x) = 1

So now we have (1-sin^2(x))/sin^2(x) and by the same identity,
1-sin^2(x) = cos^2(x), which leaves us with
cos^2(x)/sin^(x), which by known identities is cot^2(x)

Answer 5

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