Question

sec^2x/sec^2x-1=csc^2x

Answer 1

Do you need to verify the identity?

Answer 2

yes

Answer 3

Okay, first you need to change the form of the left side using fundamental trig identities. The important trig identities you need are:\[secx=\frac{ 1 }{ cosx}\]\[\sec^2x-1=\tan^2x\]

Answer 4

\[\frac{ \sec^2x }{ \sec^2x-1 }=\csc^2x\]
Since secx=1/cosx, sec^2x=1/cos^2x. Let's substitute this as well as the second identity.\[\frac{ 1 }{ \cos^2x }*\frac{ 1 }{ \tan^2x }\]

Answer 5

All we've done so far is change the form of the left side of the equation. Another important identity is \[tanx=\frac{ sinx }{ cosx }\]
We can use this identity to change the form of 1/tan^2x. Since tan^2x is on the bottom, we have to take the reciprocal of the identity. This is what we have so far. \[\frac{ 1 }{ \cos^2x }*\frac{ \cos^2x }{ \sin^2x }\]

Answer 6

The cos^2x cancel out so we're left with \[\frac{ 1 }{ \sin^2x }\] which is equal to\[\csc^2x\]. We have successfully verified the identity. When you do problems like these, make sure you use your trig identities to change the form of each side. Does that make sense?

Answer 7

yes that makes sense thank you :)

Answer 8

Of course! :)