Asked by CodyJinks

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
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

Answers

Answered by oobleck
(1+tanx)/(1+cotx) = tanx(1+tanx) / tanx(1+cotx)
= tanx(1+tanx) / (1+tanx)
= tanx
Answered by Reiny
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.
Answered by Reiny
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.
Answered by CodyJinks
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
Answered by Reiny
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
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