Question

What steps would I use to verify cot x sec^4x = cot x + 2 tan x + tan^3x?

Answer 1

you can start from right side and use
cot x = 1/ tan x
then you'll get an expression in terms of tan x and its higher powers only, the numerator factors out well...

Answer 2

if you think of tan^2 x as 'y', you'll have a quadratic expression in y in the numerator...which factors out easily.

Answer 3

oh ok then i just factor it out

Answer 4

yeah, show fist few steps what u have done ?
or should i ?

Answer 5

The only thing i dont know what to do with is the sec^4..

Answer 6

oh, thats left side
we started with right side and it will get simplified to cot x sec^4 x automatically...

Answer 7

which would make it even on both sides right?

Answer 8

you might find it easier to dispense with the trig all together for the moment
put \(\cos(x)=a\) and \(\sin(x)=b\) and start trying to show that
\[\frac{a}{a^4b}=\frac{a}{b}+\frac{2b}{a}+\frac{b^3}{a^3}\]

Answer 9

yeah, but could you simplify right side ?
what expresson in tan x did u get on right ?

Answer 10

you can simplify the left hand side right away as
\[\frac{1}{a^3b}\] and now if you add up on the right, you will get a common denominator of \(a^3b\) which is promising
then show that the numerator is 1

Answer 11

\[\frac{a}{b}+\frac{2b}{a}+\frac{b^3}{a^3}\]
\[=\frac{a^4+2a^2b^2+b^4}{a^3b}\]

Answer 12

@satellite73 how did you get those numbers on the top?

Answer 13

and the numerator is 1 because \(a^4+2a^2b^2+b^4=(a^2+b^2)^2=1^2=1\) as \(\cos^2(x)+\sin^2(x)=1\)

Answer 14

added

Answer 15

@hartnn might have a simpler way

Answer 16

oh ok so i get 1/a^3b = 1/a^3b...correct? which is 1/cos^3 sin=1/cos^3 sin

Answer 17

yes
i find it hard to work with all the trig functions when doing basic algebra

it is easier for me to work with \(a\) and \(b\) then at the end recall that \(a=\cos(x)\) and \(b=\sin(x)\) for whatever trig step is necessary

that does not mean it is the best way to do it though, just separates the basic algebra from the trig is all

Answer 18

It makes it easier for me as well but every time i try it i mess something up and i get the whole question wrong

Answer 19

you might try doing what @hartnn said, may be more straight forward
yes, the algebra is usually the hard part, the trig identities are easy and few

Answer 20

i will try! Thanks guys, Ill do my best to finish the worksheet by myself.

Answer 21

i have the same thing, just without using a,b
1/ tan x +2 tan x + tan^3 x
= (1+ 2 tan^2 x+ tan^4 x) / tan x

lets see numerator only
(1+ 2 tan^2 x+ tan^4 x)
if you look carefully, you get something of the form y^2+2y+1
which simplifies to (y+1)^2

so,
(1+ 2 tan^2 x+ tan^4 x) = (1+tan^2 x)^2
ans as we all know
1+ tan^2 x = sec^2 x
giving (1+tan^2 x)^2 = sec^4 x

Answer 22

finally,
1/ tan x +2 tan x + tan^3 x
= (1+ 2 tan^2 x+ tan^4 x) / tan x
= sec^4 x / tan x
= sec^4 x cot x

Answer 23

ask if any doubt in any step.... :)

Answer 24

Thank you @hartnn! i was always confused with the sec^4... i think i understand a bit more now... i just hate pre calc -_-

Answer 25

welcome ^_^
with some practice, i have seen many math haters turn into math lovers ;)