Question

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation.

1. 1 + sec^2x sin^2x = sec^2x

2. (sinx / 1 - cosx) + (sinx / 1 - cosx) = 2 csc x

3. - tan^2x + sec^2x = 1

It would be great to be shown step by step how to solve these problems. Thanks in advance!

Answer 1

For number one, I believe the answer is: 1 + sec^2(x) sin^2(x) = sec^2(x)

sec(x) = 1/cos(x)

sec(x)sin(x)

tan(x)

1 + tan^2(x)

sec^2(x), which equals the right side.

Please correct me if I'm wrong!!

Answer 2

Well, almost correct. sec^2(x)*sin^2(x) = tan^2(x)
But you had the correct process.

Answer 3

Ok, thanks! And for the second question I got: (sin x) / (1 - cos x) + (sin x) / (1 + cos x) = 2 csc x

(1 - cos x)(1 + cos x) (each side is then multiplied with this)

(sin x)(1+cos x) + (sin x)(1 - cos x) = 2 csc x (1 - cos^2 x)

1 - cos^2 x = sin^2 x

(sin x)(1 + cos x) + (sin x)(1 - cos x) = 2 csc x (sin^2 x)

sin x + sin x cos x + sin x - sin x cos x = 2 sin x

2 sin x

I think I did this right, because 1 / sinx equals cscx, correct?

Answer 4

You are very correct for doing that..

Answer 5

Yep, Good..

Answer 6

Wait, how you got 2isn(x)

Answer 7

*2sin(x)

Answer 8

Hm, Idk I just solved..lol is it wrong?

Answer 9

Oh ok, thanks

Answer 10

Sorry got you.. :)

Answer 11

So I'm right? :P

Answer 12

\[\frac{2\sin(x)}{\sin^2(x)} \implies \frac{2}{\sin(x)} \implies 2 cosec(x)\]

Answer 13

I see, thx

Answer 14

I think third one you can do it, right??

Answer 15

I can, I haven't done it yet though. I'll have you check my answer when I'm done, ok? :)

Answer 16

For third one, you have to do anything??

Answer 17

In first equation, you used:

\(1 + tan^2(x) = sec^2(x)\)

Just rearrange it by subtracting \(tan^2(x(\) from both the sides, you will get your desired thing..

Answer 18

So it's just 1?

Answer 19

No, its 1 = sec^2(x) - tan^2(x)

Answer 20

Oops, that's what I meant..forgot to subtract tan^2x from sec^2x lol

Answer 21

And that's the answer, right? :) Thanks for your help!

Answer 22

yes.. :)