Help with Identity proof Please

Hello, I am working on a problem in my homework where I need to prove that the identity equation is true. The parameters are, however, that I must only work on one side of the equation. The Equation:

$\displaystyle sec^2x + csc^2x = sec^2x*csc^2x$

I chose the left side to work in. First, using Pythagorean Identity, expand sec and csc:

$\displaystyle (1+tan^2x) + (1+cot^2x)$

Combine the constants:

$\displaystyle tan^2x + cot^2x + 2$

Use the co-function identities to expand again:

$\displaystyle sin^2x/cos^2x + cos^2x/sin^2x + 2/2$

Here is where I think I am stuck, or maybe I took a wrong turn before this. I could go further, and break the sin into its Pythagorean identity:

$\displaystyle (1-cos^2x)/cos^2x + cos^2x/(1-cos^2x) + 2/2$

But then what? No common denominator. I could take the reciprocal of the second term and then add, but it doesn't get me much.

$\displaystyle (1-cos^2x)/cos^2x + (1-cos^2x)/cos^2x + 2/2 $

Brings up

$\displaystyle (2-cos^2x)/cos^2x$

This equals 1, but then I still have that 2/2 to add in, so then I get 3. which is not what $\displaystyle sec^2x*csc^2x$ is.

Where have I gone wrong? (if I've posted this incorrectly, I'll edit it until I get it right. Please forgive any incorrect formatting. This is my first time here.)

Re: Help with Identity proof Please

Quote:

Originally Posted by

**DamenFaltor** Hello, I am working on a problem in my homework where I need to prove that the identity equation is true. The parameters are, however, that I must only work on one side of the equation. The Equation:

$\displaystyle sec^2x + csc^2x = sec^2x*csc^2x$

I chose the left side to work in. First, using Pythagorean Identity, expand sec and csc:

$\displaystyle (1+tan^2x) + (1+cot^2x)$

Combine the constants:

$\displaystyle tan^2x + cot^2x + 2$

Use the co-function identities to expand again:

$\displaystyle sin^2x/cos^2x + cos^2x/sin^2x + 2/2$

Here is where I think I am stuck, or maybe I took a wrong turn before this. I could go further, and break the sin into its Pythagorean identity:

$\displaystyle (1-cos^2x)/cos^2x + cos^2x/(1-cos^2x) + 2/2$

But then what? No common denominator. I could take the reciprocal of the second term and then add, but it doesn't get me much.

$\displaystyle (1-cos^2x)/cos^2x + (1-cos^2x)/cos^2x + 2/2 $

Brings up

$\displaystyle (2-cos^2x)/cos^2x$

This equals 1, but then I still have that 2/2 to add in, so then I get 3. which is not what $\displaystyle sec^2x*csc^2x$ is.

Where have I gone wrong? (if I've posted this incorrectly, I'll edit it until I get it right. Please forgive any incorrect formatting. This is my first time here.)

I started from LHS.

$\displaystyle \csc^2x\cdot \sec^2x$

$\displaystyle \csc^2x \cdot (1 + \tan^2x)$

$\displaystyle \csc^2x+\csc^2x \cdot \tan^2x$

$\displaystyle {\csc^2x+ \frac{1}{\sin^2x}} \cdot \frac{\sin^2(x)}{\cos^2(x)}$

$\displaystyle \csc^2x+\frac{1}{\cos^2x}$

$\displaystyle \csc^2x+\sec^2x$

Re: Help with Identity proof Please

Your 2 has become 2/2 = 1 between your second and third lines of working.

Multiply throughout by $\displaystyle \dfrac{\sin^2(x)\cos^2(x)}{\sin^2(x)\cos^2(x)}$

$\displaystyle = \dfrac{\sin^4(x) + \cos^4(x) + 2\sin^2(x)\cos^2(x)}{\sin^2(x)\cos^2(x)}$

The numerator will factor (it's a perfect square) to a form where you can use an identity

Re: Help with Identity proof Please

Ok I see you are going from the right hand side, the multiplication side. But.... How did you go from

$\displaystyle \csc^2x \cdot (1 + \tan^2x)$

to

$\displaystyle \csc^2x+\csc^2x \cdot \tan^2x$

Which identity is that??

Re: Help with Identity proof Please

Ack nevermind, you just multiplied through... man why didn't I see that??

Re: Help with Identity proof Please

Thank you guys! This helped me a lot.

Re: Help with Identity proof Please

Also sprach Zarathustra, This is kind of a odd question - but how did you know to pick the sec function to transform but leave the cosecant function in its existing state? Was it just a leap of intuition??

Re: Help with Identity proof Please

if you chose left hand side

sec^2 x + cosec^2 x = ( 1/cos^2 x ) + (1/sin^2 x) => [(sin^2 x) + (cos^2 x)] / (cos^2 x)(sin^2 x)

= 1 / (cos^2 x)(sin^2 x) => (sec^2 x)(cosec^2 x)

Re: Help with Identity proof Please

Quote:

Originally Posted by

**DamenFaltor** Also sprach Zarathustra, This is kind of a odd question - but how did you know to pick the sec function to transform but leave the cosecant function in its existing state? Was it just a leap of intuition??

I think that one answer to this question could be: experience that comes after solving a bunch of this kind of questions...

Re: Help with Identity proof Please

Thank you everyone for your help. I'm trying to study for a test that I have coming up next week, and I am practicing like crazy. My math anxiety is huge, so seeing how these things can get broken down into manageable pieces helps a lot.