verify the identity: 1+sec^2x/1+tan^2x=1+cos^2x

Question

verify the identity:  1+sec^2x/1+tan^2x=1+cos^2x

terenzreignz · Answer

This identity
$$\Large \frac{1+\sec^2(x)}{1+\tan^2(x)}=1+\cos^2(x)$$
?

anonymous · Answer

its easy, just plug the values of sec^2 and tan^2 in terms of sin and cos and keep on simplifying, you'll get the value on the right hand side.

terenzreignz · Answer

Or use the fact that
$$\Large 1+\tan^2(x) = \sec^2(x)$$

anonymous · Answer

good call!

anonymous · Answer

but that would be using another known identity? would it?

terenzreignz · Answer

We're supposed to know basic identities anyway.
If you like, you can always start with the pythagorean identity and work from there.
$$\Large \cos^2(x) +\sin^2(x) = 1$$
and work from there.
I avoid using the 'convert everything to sines and cosines' method to prove trigonometric identities if I can help it... it's just so... not elegant  :D
But effective.

Still prefer turning one side into the other.... it's like MAGIC! :D