Question

I need to prove cot x sec^4x = cot x + 2 tan x + tan^3x. I have a few steps down but I'm stuck.

Answer 1

So far I've started with the RHS and have 1/tanx+2tanx+tan^3x but that's it

Answer 2

ok

now change that to one fraction

= 1 + 2 Tan^2x + tan^4 x
-------------------
tan x


= ( 1 + tan^2x)( 1 + tan^2 x)
-----------------------
tan x

now sec^ x = 1 + tan^2 x

Answer 3

do you see the way to go ?

Answer 4

thats sec^2 x

Answer 5

How does 2tanx become 2tan^2x when you change it to one fraction? I get the rest of it though.

Answer 6

because you are multiplying 2 tan x by tan x

Answer 7

and tan^3 x becomes tan^4 x by the same process

Answer 8

Ohhhh okay that makes so much more sense thank you. So now would we use the tan reciprocal identity to get cot? Or do we replace 1+tan^2x with sec^2x using the Pythagorean identity?

Answer 9

replace 1 + tan^2 x by sec^2 x and use cot x = 1/ tan x
and you'll get the left hand side

Answer 10

use both

Answer 11

So wouldn't we just end up with 1/tanx * sec^4x=sec^4x

Answer 12

Wait okay no I got it. The final answer will be sec^4x/tanx. Awesome. Thank you so much!

Answer 13

yw