How would you prove or disprove #cotx - cosx/cotx = cos^2x/(1 + sinx)#?

1 Answer
Oct 30, 2016

Rewrite all terms in #cotx# as #cosx/sinx#.

#cosx/sinx - cosx/(cosx/sinx) = cos^2x/(1 + sinx)#

#cosx/sinx - cosx xx sinx/cosx = cos^2x/(1 + sinx)#

#cosx/sinx - sinx = cos^2x/(1 + sinx)#

Rewrite the right-hand side using the identity #sin^2x + cos^2x = 1#.

#cosx/sinx - sinx = (1 - sin^2x)/(1 + sinx)#

#cosx/sinx - sinx = ((1 + sinx)(1 - sinx))/(1 + sinx)#

#cosx/sinx - sinx = 1 - sinx#

#(cosx - sin^2x)/sinx = 1 - sinx#

The identity is false, because no matter what you do with the left hand side, you will never be able to get on the right.

Hopefully this helps!