Question

cot x sec4x = cot x + 2 tan x + tan3x
I have to make the left side look like the right to prove this equation is true. I've done most of it but I'm stuck at the end

cot x sec^4x = cot x + 2 tan x + tan^3x

cot x sec^2x sec^2x = cot x + 2 tan x + tan^3x

cot x + 1 + tan^2 x + 1 + tan^2 x = cot x + 2 tan x + tan^3x

Answer 1

cot x + 2 + tan^2 x +tan^2 x

Answer 2

sorry the right side has to look like the left side

Answer 3

now i'm really confused....

Answer 4

this might help you: cos2x = 2cos^2(x) -1

Answer 5

it doesnt matter which side should look like which one

Answer 6

as long as you can prove that you can derive one side from the other, your proof is correct

Answer 7

Yes, and you might also think about relevant trig identities:

cot x = (cos x) / (sin x)

sec x = 1 / (cos x)

These are just examples. You might also try tan x = (sin x) / (cos x)

Answer 8

i'd say start with the right side. it'll be easier

Answer 9

left*

Answer 10

can I just do this?
cot x + 2 + tan x + tan^3 x
cot x + 2 + tan^4 x
cot x + 2(1 + tan^2 x)
cot x + 2(sec^2 x)
cot x +sec ^4 x
which would just be cot x sec ^4 x like in the original?

Answer 11

is it sec^4x or sec4x in the left side?

Answer 12

@Helpjebal: your proposed expression shows ' 2 ' standing by itself. This is not the case in the given, original, equation. So, unfortunately, my answer is "no."

Answer 13

it's sec^4x on the left side

Answer 14

and how did you get tan^3x from tan3x?

Answer 15

oh

Answer 16

You might have to experiment (e. g., start with the left side, start with the right side) until you find yourself on the right track, especially if you're not used to this sort of problem.

Again I urge you to consider the trig identities I typed to you. So far I haven't seen you using them.

Answer 17

I didn't write it properly, it's cot x sec^4x = cot x + 2 tan x + tan^3x

Answer 18

I agree that you can work on either side, or on both sides. If you change the right side, you can always change it back to its original form in the end.

The key to proving this identity lies in the proper use fo trig identities. What is the common trig identity that shows the relationship between the cot and the sec functions?

Answer 19

Hint: this identity stems from the even more common\[\sin^2x+\cos^2x=1\]

Answer 20

okay so I think I know how to use this trig identities you showed earlier...
cot x + 2 tan x + tan^3 x
1/ tan x + 2 tan x + tan^3 x
(1 / tan x) + (2 tan x + tan^3 x / 1)
1 + 2 tan x + tan^3 x / 1 tan x

that's where I've gotten... but you said that sin^2x+cos^2x=1
so I could also do
cot x + 2 tan x + tan^3 x
cos(x)/sin(x) + 2(sin(x)/cos(x)) + tan^3x?

Answer 21

The trick here is to reduce the number of different trig functions in your expressions. Of course you can define tan x as (sin x) / (cos x), but do you really want to introduce sin x and cos x here, when the primary trig functions of the given expression are cot x and sec x? I think not.

Answer 22

Have y ou a table of trig identities available? If so, review \[\sin^2x + \cos^2x = 1\]

Answer 23

and find the identity involving \[\cot^2x\]

Answer 24

Then apply this "new" identity to simplifying the given expression. Again: seek to REDUCE (not increase) the number of trig functions you're using.

Answer 25

ok i am back

Answer 26

i did the question. there are only three steps to solving it

Answer 27

first, you want to convert the sec^4x to something with a lower order

Answer 28

use the identity sec^x = 1+tan^2x

Answer 29

sec^2x = 1+tan^2x

Answer 30

Thats how I was originally trying to solve it
cot x + 1 + tan^2x + 1 + tan^2x
cot x + 2 + tan^4 x
but I got incredibly stuck...

Answer 31

The only identity i know of involving cot^2 x is + cot^2x = csc^2x

Answer 32

Please double check before you make substitutions such as "cos x + 1." That is NOT an identity.

Answer 33

I never said cos x + 1...
2(1 + tan x) is the same as 1 + tan^2x + 1 + tan^2x thats why i wrote it like that

Answer 34

Check out physo's suggestion: sec^x = 1+tan^2x. Actually, this should be

(sec x)^2 = 1 + (tan x)^2. Is it correct? Double check. If yes, apply it to simplifying the given expression.

Answer 35

i'll give you the first step to help you:\[cotx(\sec^2x)^2\]

Answer 36

@Helpjebal :
You did type in the following:

cot x + 1 + tan^2x + 1 + tan^2x
cot x + 2 + tan^4 x

Answer 37

now use the identity on sec^x and expand it

Answer 38

sec^2x

Answer 39

Let's let physo help here. I'll be in the background if you need me.

Answer 40

okay so physo we're on the left side right?
so...
cot x sec^4x
cot x (sec^2 x)^2
cot x (1 + tan^2x)^2
am I on the right path?

Answer 41

yes

Answer 42

do you remember (a+b)^2 = a^2 + 2ab + b^2?

Answer 43

yes I remember that so...
cot x sec^4x
cot x (sec^2 x)^2
cot x (1 + tan^2x)^2
cot x + 1 + 2 tan^2 x + tan^4x

Answer 44

you need to take the brackets into consideration

Answer 45

this is fine:

cot x sec^4x
cot x (sec^2 x)^2
cot x (1 + tan^2x)^2

but the next line is not.

Answer 46

yup

Answer 47

physo is right: those parentheses / brackets are very important gatekeepers. We can't just ignore them. Anything within ( ) must be done first, followed by any exponentiation.

Answer 48

okay so
(1 + tan^2x)(1 + tan^2x)
1 + tan^2x + tan^2x + 2 tan^2x
1 + tan^4x + 2 tan^2x

making that correction it would be
cot x sec^4x
cot x (sec^2 x)^2
cot x (1 + tan^2x)^2
cot x + 1 + tan^4x + 2 tan^2x

Answer 49

wait sorry

Answer 50

i did that wrong again didn't i...

Answer 51

Please note: because that cot x is OUTSIDE your parentheses, you must multiply everything INSIDE the parentheses by cot x. Distributive rule of multiplication. ;(

Answer 52

yes mathmale is correct. you squared it correctly but you forgot to multiply it with cotx

Answer 53

ohhhhhh
cot x (1 + tan^4x + 2 tan^2x)
cot x + tan^3 + 2 tan x

Answer 54

there you go

Answer 55

Hold it, physo. That wasn't my work.

Answer 56

Thank you guys so much! I'm so incredibly thankful!

Answer 57

So glad you've been able to solve this problem!