Complete The Identity

[tristatefabricatorsinc]

Can someone please explain how to complete 1 + sin^2 x to get...

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


I am getting a little better with fundamental identities, but I am struggling with a couple of the problems I have and this is one of them.

Thank you so much for all the help everyone!

 

[soroban]

Hello, tristatefabricatorsinc!

Can someone please explain how to complete \(\displaystyle 1\,+\,\sin^2 x\) to get...

\(\displaystyle \csc^2 x\,-\,\cot^2 x\,+\,\sin^2 x\)

Are you sure you want to work in that direction?

Isn't that similar to:
\(\displaystyle \;\;\)Prove that: \(\displaystyle \,\frac{5(2^3)\,-\,2(3^2)}{4\,+\,7} \:=\;2\)
\(\displaystyle \;\;\) by starting with the \(\displaystyle 2\) and "completing" it?


Always start with the more complicated side and simplify . . .

We have: \(\displaystyle \,\frac{1}{\sin^{^2}x}\,-\,\frac{\cos^{^2}x}{\sin^{^2}x} \,+\,\sin^{^2}x\)

\(\displaystyle \;\;\;=\;\frac{1\,-\,\cos^{^2}x}{\sin^{^2}x}\,+\,\sin^{^2}x\)

\(\displaystyle \;\;\;=\;\frac{\sin^{^2}x}{\sin^{^2}x}\,+\,\sin^{^2}x\)

\(\displaystyle \;\;\;=\;1\,+\,\sin^{^2}x\)