Trig help (just sin, cos, and tan)

I need to get

tan squared x - sin squared x

to equal

tan squared x times sin squared x

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I've been trying to do this problem forever but nothing seems to be working. I feel like I just missing something obvious. Can someone please explain the steps on how you get from that first problem to the second?

Thank you!

tan²(x) - sin²(x) = tan²(x) * sin²(x)

* LHS *

First, convert tan²(x) into sin²(x)/cos²(x):

sin²(x)/cos²(x) - sin²(x)

Since the LCD is cos²(x)..

sin²(x)/cos²(x) - sin²(x) * cos²(x)/cos²(x)
= (sin²(x) - sin²(x)cos²(x))/cos²(x)

Then, factor the top expression by sin²(x):

sin²(x)/cos²(x) * (1 - cos²(x)) [Note that I factored out 1/cos²(x) just for clarity]

* * *

Recall that the Pythagorean identity is:

sin²(x) + cos²(x) = 1

Rearrange the terms:

sin²(x) = 1 - cos²(x)

So that:

sin²(x)/cos²(x) * (1 - cos²(x))
= tan²(x) * sin²(x) [Use the first-mentioned formula that tan²(x) = sin²(x)/cos²(x)]
= RHS

I hope this helps!

tan²x − sin²x
= (sin²x/cos²x) − sin²x
= [ sin²x − (sin²x)(cos²x) ] / cos²x
= [ sin²x (1−cos²x) ] / cos²x
= (sin²x)(sin²x) / cos²x
= (sin²x) [ (sin²x) / (cos²x) ]
= (sin²x) (tan²x)
= (tan²x) (sin²x).