Question

Hello, I need help with solving these trig identities. All help is appreciated!

1. Establish the Identity:
(cos(2x)/1+sin(2x)) = ((cotx-1)/(cot+1))

2. Establish the Identity
((sec²x-tan²x+tanx)/(secx)) = (sinx+cosx)

Answer 1

basic method:
test to see if it is actually true by picking a non-standard angle. That is, don't use something like 45° or 90°

usually I start with the more complicated side and try to reduce it so it becomes the other side.
#1.
LS = (cos(2x))/(1+sin(2x)) , I notice that the RS has x as the angle and not 2x, so use the formulas that reduces trig angles form 2x to x
I also corrected the brackets in your LS
= (cos^2 x - sin^2 x)/(sin^2 x + cos^2 x + 2sinxcosx), I replaced 1 with sin^2x + cos^2x
= (cosx + sinx)(cosx - sinx) / (cosx + sinx)^2
= (cosx - sinx) / (cosx + sinx)
now I have to get those cotangents in there, remember cotx = cosx/sinx
so let's divide top and bottom by sinx, realizing that sinx/sinx = 1
= (cosx/sinx - sinx/sinx) / (cosx/sinx + sinx/sinx)
= (cotx - 1) / (cotx + 1)
= RS

#2, for this one, hint: sec^2 x = 1 + tan^2 x

Answer 2

Thank you so much Reiny! That was a lot of help :)