Question

How do I simplify this expression?
cos^2 x + sin^2 x / cot^2 x - csc^2 x

Answer 1

\[\frac{ \cos^2 x + \sin^2 x }{ \cot^2 x - \csc^2 x }\]

Answer 2

Recall corresponding identities.

\[\cot x = \frac{ \cos }{ \sin }\] & \[\csc x = \frac{ 1 }{ \sin }\]

Replace the cotangent and cosecant with thereof.

\[\frac{ \cos^2x + \sin^2x }{ \frac{ \cos^2 }{ \sin^2 } - \frac{ 1 }{ \sin^2 } }\]

From there, are you able to solve for the solution?

Answer 3

the numerator = 1

Answer 4

^ Yes. It's a Pythagorean identity.

Answer 5

so that leaves it as \[\frac{ \cos^2 x + \sin^2 x }{ \frac{ \cos^2 }{ \sin^2 } }\]
right? where do I go from there?

Answer 6

ohhhh sin^2 x + cos^2 x = 1 is a pythagreon identity but what do we do with the denominator?

Answer 7

1/(cot^2) = tan^2(x)

Answer 8

because cos^2 + sin^2 = 1 and cos^2/sin^2 = cot^2

Answer 9

I'm finding this really hard to understand... I'm sorry

Answer 10

cos^2 x 1
------ - ---
sin^ 2 x sin^2 x

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

Answer 11

= - sin^2 x
-------- = -1
sin^2 x

Answer 12

so we end up with 1 / -1 = -1

Answer 13

how did -sin^2x end up in the numerator?

Answer 14

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

Answer 15

- subtract sin^2 x and 1 from both sides

Answer 16

ohhhh I see! thank you!!

Answer 17

yw