Prove - tan^2(x) + sec^2(x) = 1 by working on one side to match the other using identities.

Question

Prove  - tan^2(x) + sec^2(x) = 1 by working on one side to match the other using identities.

anonymous · Answer

$$- \tan^2x + \sec^2x = 1$$

anonymous · Answer

$$- \tan^2x + \sec^2x = 1$$

nnesha · Answer

sec^2 theta = what ?
remember the identity ?

anonymous · Answer

1/cos^2x

nnesha · Answer

well that's reciprocal of sec but it's okay we can use that too!!
tan^2 =what ?

anonymous · Answer

sin^2x/cos^2x

anonymous · Answer

or 1/cot^2x

nnesha · Answer

yes right so replace tan and sec with that $$\huge\rm -\frac{ \sin^2x }{ \cos^2x } +\frac{ 1 }{ \cos^2x}$$
find the common denominator

anonymous · Answer

cos^2x?

nnesha · Answer

ohh well not gonna work should use the identity i guess

jhannybean · Answer

You could also use the fact that $\sf sec^2(\theta) = tan^2(\theta) +1$ and then substitute this in place of $\sf \sec^2(\theta)$

anonymous · Answer

thats true, thanks

jhannybean · Answer

$$\sf -tan^2(\theta)+\sec^2(\theta) = 1$$$$\sf -\tan^2(\theta) +\color{red}{\tan^2(\theta) +1}=1$$

anonymous · Answer

Wow... The one identity I didn't think of solved it so easily. Thank you!

nnesha · Answer

$$\huge\rm \frac{ -\sin^2x +1}{ \cos^2 }$$
use the special identity sin^2x+cos^2x =1 solve for cos^2

jhannybean · Answer

No problem :)

nnesha · Answer

here you can copy these identities http://www.math.com/tables/trig/identities.htm you weren't familiar with this so that's why i thought better to write interms of sin and cos

jhannybean · Answer

That's a good way to approach it too. @Nnesha :)

anonymous · Answer

Thanks that will help too @Nnesha

nnesha · Answer

yw :=)