How does one verify (secx-cosx)^2=tan^2x-sin^2x?

(secx-cosx)^2=tan^2x-sin^2x

1 Answer
Apr 23, 2018

See below.

Explanation:

We'll leave the right side alone and attempt to get the left side to match it.

Expand the left side:

(secx-cosx)(secx-cosx)=tan^2x-sin^2x

sec^2x-2secxcosx+cos^2x=tan^2x-sin^2x

2secxcosx=2/cancelcosxcancelcosx=2, so we get

sec^2x-2+cos^2x=tan^2x-sin^2x

Rewrite, splitting up the -2 into -1-1:

(sec^2x-1)+(cos^2x-1)=tan^2x-sin^2x

Recall the following identities:

sec^2x-1=tan^2x
sin^2x+cos^2x=1 ->cos^2x-1=-sin^2x

So, we get

tan^2x-sin^2x=tan^2x-sin^2x