Proving identity

Mar 2017
358
3
Massachusetts
Hi,

Does the following expression represent an identity?

19718665_1460132910709465_697865054_o.jpg


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:

19718645_1460131864042903_41735955_o.jpg

Look forward to a response! Any help is appreciated!
 

Plato

MHF Helper
Aug 2006
22,506
8,663
$ \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*}$
 
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Mar 2017
358
3
Massachusetts
Great, thanks for your help. I understand what I missed.