Question

Here is one I have absolutely no idea how to work. I'm clueless here.

verify the identity:

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

Answer 1

So with an identity, we want to try and manipulate one side, to make it look like the other. So let's leave the secant function alone, and try to shrink that giant mess down to match the right side.

Answer 2

Luckily all of our trig functions are given in terms of X, not 2x or 3x.. so we won't have to worry about any double/half angle shinanigans! :D

Answer 3

Getting a common denominator between the fractions gives us...
\[\left(\frac{ \cos x }{ \cos x }\right) \frac{ \cos x }{ 1 + \sin x } + \frac{ 1+\sin x }{ \cos x }\left(\frac{ 1+\sin x }{ 1+ \sin x }\right)\]

Answer 4

\[\frac{ \cos^2 x }{ \cos x(1+ \sin x) }+\frac{ (1+\sin x)^2 }{ \cos x (1+ \sin x) }\]

Answer 5

\[\frac{ \cos^2 x + 1 + 2\sin x + \sin^2 x }{ \cos x (1+\sin x) }\]

Answer 6

recalling that cos^2 + sin^2 gives us 1, this simplifies to:
\[\frac{ 2+2\sin x }{ \cos x (1+\sin x) }\]

Answer 7

(cosx/1 + sinx) + (cosx/1-sinx)=2sec

[cosx(1-sinx)+cosx(1+sinx)]/(1-sin^2x)=2sec x

(cosx-sinxcosx+cosx+sinxcosx)/cos^2x=2sec x

2cosx/cos^2x=2sec x

2/cosx=2sec x

2secx=2secx

Answer 8

\[\frac{ 2(1+\sin x) }{ \cos x (1 + \sin x) }\]

Answer 9

\[\large \frac{ 2 }{ \cos x }=2\sec x\]

Answer 10

This one is a bit of a doozy darian :C lot of steps to get through it, confused about anything in there? :O

Answer 11

Dragon used an alternative method that you could try also c:
With these trig functions there are often maybe paths you can take to get them to match up, since there are so many trig identities! :D