For this question, they want me to use fundamental trig identities to simplify the expression. The problem is as follows; (tanx/csc^2x + tanx/sec^2x)(1+tanx/1+cotx) - 1/cos^2x

Question

I got as far as this; tanx(1/csc^2x + 1/sec^2x)(1+tanx/1+cotx) - sec^2x. I factored out the tangent and simplified the 1/cos^2x to sec^2x. Then I simplified further by saying that tanx(sin^2x+cos^2x)((1+tanx/1+cotx)-sec^2x. just not sure how to simplify down the 1+tanx/1+cotx. Some help would be much obliged

Answer 1

(1+tanx)/(1+cotx) = tanx(1+tanx) / tanx(1+cotx)
= tanx(1+tanx) / (1+tanx)
= tanx

Answer 2

I am going to insert some necessary brackets where I think they probably should be:
tanx(1/csc^2x + 1/sec^2x)((1+tanx)/(1+cotx)) - sec^2x
= tanx(sin^2 x + cos^2 x)(1+tanx)/(1+cotx) - (tan^2 x + 1)
= tanx (1)(1+tanx)/(1+cotx) - tan^2x - 1

check your typing, this does not reduce to the answer you stated.

Answer 3

Ok, picking up from oobleck's
(1+tanx)/(1+cotx) = tanx, we get

tanx (tanx) - tan^2 x -1
= -1 , which would not be the answer you gave.

Answer 4

i never gave a final answer just as far as i got.

oobleck are you able to explain what identities you used to break down that part? I'm just not seeing how it was simpified

Answer 5

looks like oobleck is not online, so I will explain

As he has shown, he has multiplied top and bottom by tan x
(1+tanx)/(1+cotx) = tanx(1+tanx) / tanx(1+cotx)
left the top as is, but expanded the bottom, realize that tanxcotx = 1
= tanx(1+tanx) / (1+tanx)

so the bottom becomes tanx + 1, cancels the top 1+tanx, leaving tanx