Question

Prove that the given equations are identities: (1+tan(theta/2))/(1-tan(theta/2)) = tan(theta)+sec(theta)

They can be proven, but the book doesn't show how... :(

Answer 1

convert everything into sin and cos and work from there tell me if you run into any problems ill guide you to the answer :)

Answer 2

I've actually tried a lot of things... I always just make a mess on my paper :(

If you convert the problem into sin and cos, you get: (1+[(sin(theta)/(1+cos(theta))])/(1-[(sin(theta)/(1+cos(theta))])=tan(theta)+sec(theta)

Answer 3

i am going to make theta=x ok just for simplicity ok so first lets start from the right side okay so we got sinx/cosx + 1/cosx = (sinx +1)/cosx

so lets try doing the left side and try to get here somehow alright

Answer 4

so now we got (1+tan(x/2))/(1-tan(x/2)) for the numerator we have [cos(x/2) + sin(x/2)]/cos(x/2) and now for the denominator we got [cos(x/2) - sin(x/2)]/cos(x/2)

and now if we divide it by each other we will get [cos(x/2) + sin(x/2)]/cos(x/2) TIMES cos(x/2)/[cos(x/2) - sin(x/2)] notice cos(x/2) cancels out and now we are left with

[cos(x/2) + sin(x/2)]/[cos(x/2) - sin(x/2)]

Answer 5

hmm now what

Answer 6

cos(theta/2) has an identity that is equal to + or - RADICAL----> ((1+cos(theta)/2) if that helps

Answer 7

Though when I used that identity things got complicated...

Answer 8

if we multiply [cos(x/2) + sin(x/2)] by the top and the bottom we will get [cos(x/2) + sin(x/2)]^2/cosx

Answer 9

almost there buddy

Answer 10

do what i did multiply that at the top and by the bottom

Answer 11

and for the bottom use the identity (cosx)^2 - sinx^2 = cos2x

Answer 12

in this case x = x/2 so when we put that in cos 2x we just get cos x alright

Answer 13

OKK GOT IT

Answer 14

[cos(x/2) + sin(x/2)]^2 alright now if we expand this we get cos(x/2)^2 + 2cos(x/2)sin(x/2) + sin(x/2)^2

and now use the identity cosx^2 + sinx^2 = 1

so in this case we got 1 + 2cos(x/2)sin(x/2

and now use the identity that sin2x = 2sinxcosx

in this case our x=x/2 and if we put it into sin2x we get sinx

SOO therefore we got (1 + sinx)/cosx just as we wanted :)

Answer 15

fully understood?

Answer 16

I think so. That is a lot of work @_@

Answer 17

yeahh you basically had to use 3 identities and a lot of thinking

Answer 18

try to study over it

Answer 19

Alright, got it!

Answer 20

thats enough bro just helping you out :)

Answer 21

Is there anyway I can reward you more? I'm new to this website and gave you best answer but don't know how to give you more rewards other than that.