Completing Identities

[tristatefabricatorsinc]

Basically I have been given a list of identities and it says to complete them and I have a bunch of choices ina column, anyways, I do not know how they came up with the answers they did in the book... for example below...


A) I do not understand how sec^2 x - 1 can become sin^2x / cos^2 x

B) I do not understand how my book gets...

1 + sin^2 x = csc^2 x - cot^2 x + sin^2 x

C) How would you complete sec x / csc x

D) How would you complete cos^2 x

Is there any way someone could explain for my questions A & B how it came up with what I specified and for C & D how you would complete the identities? I am very confused on this..

Thanks a Bunch!!

 

[Gene]

Usually the easiest way is to make everything into sine and cosines.
sec^2 (x) - 1 =
1/cos^2 (x) - 1 =
1/cos^2 (x) - cos^2(x)/cos^2(x) =
(1-cos^2(x))/cos^2(x) =
sin^2(x)/cos^2(x)
If you understand what I did, try the same thing with B.
For the rest we need the whole question.

 

[soroban]

Hello, tristatefabricatorsinc!

A) I do not understand how $\displaystyle \sec^2 x\,-\,1$ can become $\displaystyle \frac{\sin^2x}{\cos^2x}$

You're expected to know the identity: $\displaystyle \,\sec^2x\,-\,1\:=\:\tan^2x$

Then: $\displaystyle \;\tan^2x\:=\:\left(\frac{\sin x}{\cos x}\right)^2\:=\:\frac{\sin^2x}{\cos^2x}$

B) I do not understand how my book gets: $\displaystyle \,1\,+\,sin^2x\:=\:\csc^2x\,-\,\cot^2x\,+\,\sin^2x$

You're also expected to know: $\displaystyle \,\csc^2x\:=\:\cot^2x\,+\,1$

Then the right side is: $\displaystyle \,(\cot^2x\,+\,1)\,-\,\cot^2x\,+\,\sin^2x\;=\;1\,+\,\sin^2x$

C) How would you complete $\displaystyle \frac{\sec x}{\csc x}$

Not sure what "complete" means here . . .

We have: \(\displaystyle \L\,\frac{\frac{1}{\cos x}}{\frac{1}{\sin x}}\)\(\displaystyle \:=\:\L\frac{\sin x}{\cos x}\)$\displaystyle \:=\:\tan x$

D) How would you complete $\displaystyle \cos^2x$

This can go in a lot of directions.

But they probably expect: $\displaystyle \,\cos^2x\;=\;1\,-\,\sin^2x$