# Proving identity

#### otownsend

Hi,

Does the following expression represent an identity?

A student of mine created the expression above and believes that it is an identity. However, I subbed in pi/4 for x on both sides and I got different answers... therefore disproving it as an identity. Can someone please confirm this with me?

I also made an attempt to solve the left-side, but didn't go very far because I didn't see much light at the end of the tunnel towards proving the expression:

Look forward to a response! Any help is appreciated!

#### Plato

MHF Helper
\begin{align*}\tan^4(x)+\tan^2(x)+1&=\dfrac{\sin^4(x)}{\cos^4(x)}+\dfrac{\sin^2(x)}{\cos^2(x)}+1 \\&=\dfrac{\sin^4(x)+\sin^2(x)\cos^2(x)+\cos^4(x)}{\cos^4(x)}\\&=\dfrac{(\sin^2(x)+\cos^2(x))^2-\sin^2(x)\cos^2(x)}{\cos^4(x)}\\&=\dfrac{1-\sin^2(x)\cos^2(x)}{\cos^4(x)} \end{align*}

1 person

#### otownsend

Great, thanks for your help. I understand what I missed.