Question 433559
use these identities:



tan^2x+1 = sec^2x)


1 + cot^2x = csc^2x


prove that left side is equal to right side:

sec^2x-csc^2x=(tan^2x + 1)-(1 +cot^2x 


=tan^2 x + 1-1 - cot^2 x

=tan^2x - cot^2x

=(tan x - cotx)(tan x + cotx)..................use identities tanx= sinx/cosx  and cotx = cosx/sinx

=(tan x - cotx)(sin x/cos x +cos x/sin x)

=(tan x - cotx)((sin xsin x +cos xcos x)/sin xcos x

=(tan x - cotx)((sin ^2x +cos^2 x)/sin xcos x

={{{(tan x - cotx)(1/sin xcos x)}}}

={{{(tan x - cotx)/sin xcos x}}}