Help with trig proof identities c3 Please

Watch
MrToodles4
Badges: 10
Rep:
?
#1
Report Thread starter 3 years ago
#1
Prove the follow:

(Sec^2 x + tan^2 x) (cosec^2 x + cot^2 x) = 1 + 2sec^2 xcosec^2 x

So I've replaced tan squared x and cot squared x with their identities (e.g. sec squared x minus 1). And then i've expanded the brackets but I get:

4sec^2 xcosec x - 2sec^2 x - 2cosec^2 x + 1

Which isn't the identity I had to get to and Im not sure what to do from here. Any help would be greatly appreciated - thanks.
0
reply
RDKGames
Badges: 20
Rep:
?
#2
Report 3 years ago
#2
(Original post by MrToodles4)
Prove the follow:

(Sec^2 x + tan^2 x) (cosec^2 x + cot^2 x) = 1 + 2sec^2 xcosec^2 x

So I've replaced tan squared x and cot squared x with their identities (e.g. sec squared x minus 1). And then i've expanded the brackets but I get:

4sec^2 xcosec x - 2sec^2 x - 2cosec^2 x + 1

Which isn't the identity I had to get to and Im not sure what to do from here. Any help would be greatly appreciated - thanks.
How about you just express the first and second brackets as fractions in terms of sin and cons, and THEN multiply them together, and then simplify?
0
reply
RDKGames
Badges: 20
Rep:
?
#3
Report 3 years ago
#3
Otherwise, you need to convince yourself that -2 \sec^2(x) - 2\mathrm{cosec}^2(x) \equiv -2 \sec^2(x)\mathrm{cosec}^2(x)
0
reply
davros
  • Study Helper
Badges: 16
Rep:
?
#4
Report 3 years ago
#4
(Original post by MrToodles4)
Prove the follow:

(Sec^2 x + tan^2 x) (cosec^2 x + cot^2 x) = 1 + 2sec^2 xcosec^2 x

So I've replaced tan squared x and cot squared x with their identities (e.g. sec squared x minus 1). And then i've expanded the brackets but I get:

4sec^2 xcosec x - 2sec^2 x - 2cosec^2 x + 1

Which isn't the identity I had to get to and Im not sure what to do from here. Any help would be greatly appreciated - thanks.
There are numerous ways to start this.

Personally, I would just multiply out the brackets first and look for a couple of terms that are what you want to see on the RHS. There are a couple of terms left that aren't quite what you want, so you need to try to manipulate them into the correct format...
0
reply
MrToodles4
Badges: 10
Rep:
?
#5
Report Thread starter 3 years ago
#5
(Original post by RDKGames)
Otherwise, you need to convince yourself that -2 \sec^2(x) - 2\mathrm{cosec}^2(x) \equiv -2 \sec^2(x)\mathrm{cosec}^2(x)
Sorry, what do you mean exactly here?
0
reply
MrToodles4
Badges: 10
Rep:
?
#6
Report Thread starter 3 years ago
#6
(Original post by davros)
There are numerous ways to start this.

Personally, I would just multiply out the brackets first and look for a couple of terms that are what you want to see on the RHS. There are a couple of terms left that aren't quite what you want, so you need to try to manipulate them into the correct format...
Yeah I've tried but Im not sure how to
0
reply
MR1999
Badges: 21
Rep:
?
#7
Report 3 years ago
#7
(Original post by MrToodles4)
Yeah I've tried but Im not sure how to
Use  \tan \equiv \dfrac{\sin}{\cos}, \cot \equiv \dfrac{\cos}{\sin} , \sec \equiv \dfrac{1}{\cos}, \csc \equiv \dfrac{1}{\sin} and rewrite the expression.
0
reply
MrToodles4
Badges: 10
Rep:
?
#8
Report Thread starter 3 years ago
#8
(Original post by Desmos)
Use  \tan \equiv \dfrac{\sin}{\cos}, \cot \equiv \dfrac{\cos}{\sin} , \sec \equiv \dfrac{1}{\cos}, \csc \equiv \dfrac{1}{\sin} and rewrite the expression.
Yeah I've been using this - still not getting anywhere sadly
0
reply
RDKGames
Badges: 20
Rep:
?
#9
Report 3 years ago
#9
(Original post by MrToodles4)
Sorry, what do you mean exactly here?
Clearly what you've done is correct, but for your expression to be equivalent to 1 + 2 \sec^2(x) \mathrm{cosec}^2(x) then it is only possible if -2 \sec^2(x) - 2\mathrm{cosec}^2(x) \equiv -2 \sec^2(x)\mathrm{cosec}^2(x)

Once you've proven this, then you can finish it off in your original working.
0
reply
davros
  • Study Helper
Badges: 16
Rep:
?
#10
Report 3 years ago
#10
(Original post by MrToodles4)
Yeah I've tried but Im not sure how to
Do you agree that if you multiply out you get this:

sec^2x cosec^2x + sec^2x cot^2x + tan^2x cosec^2x + tan^2x cot^2x

You want to keep hold of the 1st term because it's going to contribute to the RHS you're trying to get to, and you should be able to simplify the 4th term immediately from the definitions of tan and cot.

So what can you do with the 2nd and 3rd terms?
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

Current uni students - are you thinking of dropping out of university?

Yes, I'm seriously considering dropping out (18)
18.56%
I'm not sure (2)
2.06%
No, I'm going to stick it out for now (28)
28.87%
I have already dropped out (3)
3.09%
I'm not a current university student (46)
47.42%

Watched Threads

View All
Latest
My Feed