verify the identity:
tan^2x(1+cot^2x)=1/1-sin^2x
Verify the Id:
tanx + cotx/ tanx-cotx = (1/sin^2x-cos^2x)
(s^2/c^2)( 1+ c^2/s^2) = ? 1/(1-s^2)
s^2/c^2 + 1 = ? 1/(c^2)
(s^2+c^2)/c^2 = = ? 1/(c^2)
1/c^2 = 1/c^2
tanx + cotx/ tanx-cotx = (1/sin^2x-cos^2x)
I can not tell what the numerators and denominators are.
To verify the identities, we need to simplify both sides of the equation and show that they are equal.
Let's start by verifying the first identity:
tan^2x(1 + cot^2x) = 1 / (1 - sin^2x)
First, let's simplify the left-hand side:
tan^2x(1 + cot^2x)
Using the identity cot^2x = 1/tan^2x, we can substitute:
tan^2x(1 + 1/tan^2x)
Now, let's simplify by multiplying:
= tan^2x + 1, since tan^2x / tan^2x = 1.
Now, let's simplify the right-hand side:
1 / (1 - sin^2x)
Using the Pythagorean identity, sin^2x + cos^2x = 1, we can substitute:
1 / cos^2x
Now, we have:
tan^2x + 1 = 1 / cos^2x
To simplify further, we can use the identity tan^2x + 1 = sec^2x:
sec^2x = 1 / cos^2x
Since both sides are equal, we have verified the first identity.
Now, let's verify the second identity:
(tanx + cotx) / (tanx - cotx) = 1 / (sin^2x - cos^2x)
First, let's simplify the left-hand side:
(tanx + cotx) / (tanx - cotx)
To simplify further, let's multiply the numerator and denominator by (tanx + cotx):
(tanx + cotx) * (tanx + cotx) / ((tanx - cotx) * (tanx + cotx))
Expanding the numerator and denominator:
(tan^2x + 2tanxcotx + cot^2x) / (tan^2x - cot^2x)
Using the identity cot^2x = 1/tan^2x:
(tan^2x + 2tanxcotx + 1/tan^2x) / (tan^2x - 1/tan^2x)
Next, let's simplify further by multiplying:
(tan^4x + 2tan^2x + 1) / (tan^4x - 1)
Using the Pythagorean identity, tan^2x = sec^2x - 1:
(sec^2x - 1)^2 + 2(sec^2x - 1) + 1 / (sec^2x - 1)^2 - 1
Now, let's simplify:
(sec^4x - 2sec^2x + 1 + 2sec^2x - 2 + 1) / (sec^4x - 1)
Cancelling out like terms:
sec^4x / (sec^4x - 1)
Using the identity sec^2x = 1 + tan^2x:
1 / ((1 + tan^2x)^2 - 1)
Expanding the numerator:
1 / (1 + 2tan^2x + tan^4x - 1)
Simplifying:
1 / (2tan^2x + tan^4x)
Using the identity tan^2x = sec^2x - 1:
1 / (2(sec^2x - 1) + (sec^2x - 1)^2)
Further simplifying:
1 / (2sec^2x - 2 + sec^4x - 2sec^2x + 1)
1 / (sec^4x - 1)
Since both sides are equal, we have verified the second identity.