cot x sec^4 (x) = cot x + 2 tan x + tan^3 (x)
how do i make these equal and please show all the steps because i want to know how to do this

Question

cot x sec^4 (x) = cot x + 2 tan x + tan^3 (x)
how do i make these equal and please show all the steps because i want to know how to do this

larseighner · Answer

What is sec^2 equal to?

anonymous · Answer

isnt it 1+tan^2x

larseighner · Answer

Right. So substitute that twice for sec^4 x.

anonymous · Answer

ok

larseighner · Answer

In other words you have (1 + tan^2s)^2 (and you still have to mutiply that by cot x; So expand (1+ tan^2)^2

anonymous · Answer

so it would be (cotx)(1+tan^2x)(1+tan^2x)

larseighner · Answer

Right, but multiply the 1+ tan^2x terms by each other.  If you cannot figure that out, just expand (1+a)^2 and the substitute tan^2 for a when you are done.

anonymous · Answer

ok now what

larseighner · Answer

Then multiply by the cot x.  And that should give you the answer.

anonymous · Answer

? im confused

larseighner · Answer

How so?  Do you not know what (cot x)(tan x) is?

anonymous · Answer

well this is what i have now
(cotx)(1+tan^2x)(1+tan^2x)=cotx+2tanx+tan^3x

larseighner · Answer

Isn't that what you were trying to prove?

anonymous · Answer

yes

larseighner · Answer

Well, you've done it.

anonymous · Answer

i dont see how

larseighner · Answer

$$ \large \begin{align} &\cot x \sec^4 (x) \cr &=  (\cot x)(1+\tan^2x)(1+\tan^2x) \cr  &=  (\cot x)(1+2\tan^2x +\tan^4x) \ \cr  &=  \cot x+2\tan x +\tan^3x \end{align}$$

That is what you did.

larseighner · Answer

You took sec^4 x to be sec^2 x sec^2x,  You substituted (1 + tan^2x) for each occurrence of sec^2 x.  Then you multiplied the terms (1 + tan^2).  Then you multiplied by cot x.  Since cot x is the reciprocal of tan x, cot x canceled one tan x out of each term with a tan x.

anonymous · Answer

ohhh ok i get it now

anonymous · Answer

thanks