(1-sin^2x)(1+tan^2x) = 1?

Question

Can anyone prove this identity?

is this correct?

(cos^2x)(sec^2) = 1
1= 1

THANKS FOR YOUR HELP
<3

Answer 1

Yes you're correct. One line of working.

Answer 2

You have the idea perfectly, but I wouldn't set it out like that.
Write:
(1-sin^2x)(1+tan^2x)
= ( cos^2(x) )( sec^2(x) )
= 1.
That shows that you started with the left hand side, manipulated it, and ended up with the right hand side. If you start off as you have done, you are assuming the identity is true before you've proved it, and end up proving 1 = 1.

When proving identities, you should either:
(a) start with one side only, and manipulate to produce the other side; or
(b) manipulate one side until you can go no further. Then start from the other side, and manipulate that until you get the same result.

Answer 3

Let's give it a try :)

Because sin^2 + cos^2 = 1, the first factor is
... 1 - sin^2 x = cos^2 x
For the second factor,
... 1 + tan^2 x = 1 + (sin^2 x)/(cos^2 x)
... ... = (cos^2 x + sin^2 x)/(cos^2 x)
... ... = 1/cos^2 x.
Multiply them, you find
... (cos^2 x) (1/cos^2 x) = 1.

It looks like your proof is correct, but of course you would have to show that sec^2 x = 1 + tan^2 x.

Answer 4

yes that is correct. Since (1-sin^2x) is equal to cos^2x and (1+tan^2x) is equal to sec^2, then their product (since sec is the inverse of cosine) is equal to 1.

Answer 5

You are correct.

Answer 6

That is exactly correct.

Answer 7

You got it!