Question

Verify the identity.
((cos x)/1+sin x) + ((1+sinx)/cos x) =2secx

Answer 1

\[\frac{ cosx }{ 1+\sin x }+\frac{ 1+\sin x }{ \cos x }=2secx\]

Answer 2

I don't know if I'm doing this right. I used LCD.

Answer 3

\[LEFT SIDE: \frac{ \cos^2x +(1+\sin x)(1+\sin x)}{ \cos(1+sinx) }\]

Answer 4

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

Answer 5

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

Answer 6

change cos^2 by 1-sin^2x identity

Answer 7

Wait, am I doing it right?

Answer 8

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

Answer 9

@Mertsj What did you do first?

Answer 10

Multiply first fraction by (1-sinx)/(1-sinx)

Answer 11

Multiply second fraction by cosx/cosx

Answer 12

Ohhh is it like multiplying the conjugate?

Answer 13

it's really setting up cos^2x in both denominators

Answer 14

Because 1-sin^2x=cos^2x

Answer 15

\[\huge\rm \frac{ \color{Red}{\cos^2x} +1 +2sinx + \sin^2 }{ cosx(1+sinx) }\]
\[\large\rm \frac{ \color{Red}{1-sin^2x} +1 +2sinx + \sin^2 }{ cosx(1+sinx) }\]
sin^2 cancel each other out
you will get
\[\large\rm \frac{ \color{Red}{1\cancel{-sin^2x}} +1 +2sinx +\cancel{ \sin^2} }{ cosx(1+sinx) }\]

Answer 16

Ohhh so what I was doing is wrong. Haha! Thanks!

Answer 17

i don't think so..

Answer 18

Not wrong, such not very effective.l

Answer 19

Like it the solution would be longer? :D

Answer 20

\[\huge\rm \frac{ 2+2sinx }{ cosx(1+sinx) }\]
take out 2 \[\rm \frac{ 2(1+sinx) }{ cosx(1+sinx)}\]

Answer 21

There's no "it" in my last sentence. Haha.

Answer 22

Typically there are a variety of ways to solve these identities.

Answer 23

Ohhhh @Nnesha thank you so much! :) You did the whole solution for me.

Answer 24

true but i think the way you were doing is easy :D

Answer 25

That's why this is harder because there are so many ways to do it and it's so frustrating haha!

Answer 26

and like mertsj said there are more than 2 ways to verify the identities