Solving for cot2x + sec2x = tan2x + csc2x

[Veronica_Oles]

Homework Statement

cot²x + sec²x = tan²x + csc²x

Homework Equations

The Attempt at a Solution

I began by working on my left side.

I got up until
=cos^4x + sin^2x / sin^2x cos^2x

And unsure of where to go next.

 

[SteamKing]

Homework Statement

cot²x + sec²x = tan²x + csc²x

Homework Equations

The Attempt at a Solution

I began by working on my left side.

I got up until
=cos^4x + sin^2x / sin^2x cos^2x

And unsure of where to go next.

You made a good start working on the LHS. Do the same to the RHS and see what you get.

 

[Veronica_Oles]

You made a good start working on the LHS. Do the same to the RHS and see what you get.

On right side I got sin^4x + cos^2x / cos^2x sin^2x , I can't seem to get it

 

[SteamKing]

On right side I got sin^4x + cos^2x / cos^2x sin^2x , I can't seem to get it

Compare the LHS to the RHS now. Do you notice anything in common?

 

[Veronica_Oles]

Compare the LHS to the RHS now. Do you notice anything in common?

L.S = cos^4x / sin^2x cos^2x + sin^2x / sin^2x cos^2x
= cos^2x / sin^2x + 1 / cos^2x
= cos^2x + sin^2x / sin^2x cos^2x
= 1 / sin^2x cos^2x

R.S = sin^4x / sin^2x cos^2x + cos^2x / sin^2x cos^2x
= sin^2x / cos^2x + 1 / sin^2x
= sin^2x + cos^2x / sin^2x cos^2x
= 1 / sin^2x cos^2x
Would this be the answer?

 

[SteamKing]

L.S = cos^4x / sin^2x cos^2x + sin^2x / sin^2x cos^2x
= cos^2x / sin^2x + 1 / cos^2x

I'm having a problem seeing how you go from the expression above to the one below:

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

I see that you found the common denominator, but your numerator is incorrect.

= 1 / sin^2x cos^2x

R.S = sin^4x / sin^2x cos^2x + cos^2x / sin^2x cos^2x
= sin^2x / cos^2x + 1 / sin^2x
= sin^2x + cos^2x / sin^2x cos^2x
= 1 / sin^2x cos^2x
Would this be the answer?

Same comments from the LHS calculations apply above.

Compare the expression you found on the RHS here with the one you obtained in Post #3:

On right side I got sin^4x + cos^2x / cos^2x sin^2x

 

[SammyS]

L.S = cos^4x / (sin^2x cos^2x) + sin^2x / (sin^2x cos^2x)
= cos^2x / sin^2x + 1 / cos^2x
= ( ? × cos^2x + sin^2x) / (sin^2x cos^2x)
= 1 / (sin^2x cos^2x)

R.S = sin^4x / (sin^2x cos^2x) + cos^2x / (sin^2x cos^2x)
= sin^2x / cos^2x + 1 / sin^2x ( No idea what you did here. Whatever, it's not legal.)
= (sin^2x + cos^2x) / (sin^2x cos^2x)
= 1 / (sin^2x cos^2x)
Would this be the answer?

When writing "fractions" with an "in-line" format, one using the " / " character, you need to use parentheses to include (the entire numerator) / (the entire denominator).

I have added parentheses to what I assume are the proper locations in the above included quote. (Also added a few other items.)