Question

((Cos x)/(1+sinx)) + ((1+sinx)/(cosx)) = 2 sec x. Verify the identity

Answer 1

@mathstudent55

Answer 2

combine fractions what do you get?

Answer 3

2 cosx + 4 sinx= 2 sec x ?

Answer 4

no

what is common denominator

Answer 5

Im not sure. X

Answer 6

ok

\[\frac{1}{x} + \frac{2}{x-1}\]
how would you add these fractions?

Answer 7

X(x-1)

Answer 8

good, and what would the numerator look like

Answer 9

Or 1x+(2x-2)

Answer 10

not quite, you multiply numerators by missing factor so

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

Answer 11

Oh okày

Answer 12

now apply to trig fractions ... what is common denominator

Answer 13

Okay please give me a second

Answer 14

I dont see how will you get it by adding the fractions @dumbcow

Answer 15

i would rather use
\((1-a^2)=(1-a)(1+a)\)

:)

Answer 16

@myko , you do after simplifying and using (sin^2 +cos^2=1)
also the "1+sin" cancels out

Answer 17

Cosx(cosx)+((1+sinx)(1+sinx)) / (1+sinx)(cosx)

Answer 18

very good

Answer 19

now multiply out the numerator and we will try to combine some terms
remember to FOIL the (1+sin)(1+sin)

Answer 20

Cosx^2 +(1+sinx)^2 /

Answer 21

Isn't the denominator (1+sinx)(cosx)

Answer 22

yes

Answer 23

ok but what is (1+sin)^2 ... multiply it out

Answer 24

(1+sin)(1+sin)

Answer 25

FOIL or distribute

(x+1)(x+3) = x^2 +4x+3

Answer 26

1+ 2sin+ sin^2

Answer 27

yes good

now there is the pythagorean identity you should have learned that says:
sin^2 + cos^2 = 1

Answer 28

I'm getting confused with so many steps :/

Answer 29

hmm ok i will write it out
\[\frac{\cos^2 x + (1+\sin x)(1+\sin x)}{\cos x (1+\sin x)}\]
\[= \frac{\cos^2 x + \sin^2 x + 2 \sin x + 1}{\cos x (1+\sin x)}\]
\[= \frac{1 +2 \sin x +1}{\cos x(1+\sin x)}\]
\[= \frac{2 +2 \sin x}{\cos x (1+\sin x)}\]
\[= \frac{2(1 + \sin x)}{\cos x (1+\sin x)}\]
\[= \frac{2}{\cos x}\]
\[= 2 \sec x\]

Answer 30

In the third step how did you get rid of cos?

Answer 31

by using the identity i mentioned

sin^2 + cos^2 = 1

whenever you see "sin^2 + cos^2" you can replace it with "1"

Answer 32

http://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity

Answer 33

Oh okay I see

Answer 34

I understand now that I see it in steps. So in the end I'm left with 2secx=2secx

Answer 35

yep, when verifying an identity once you show that right side = left side. you are done

Answer 36

Oh okay and what if the sides do not equal?

Answer 37

then you have more work to do :)

depending on the problem, it may be easier to start on right side or even change both sides so that you can show right side = left side

Answer 38

Oh okay thanks soooo much for explaining it all in detail!

Answer 39

yw:)