Question

By writing (cos x) in terms of (1/2)x, find an alternative expressions for \[{1-cosx \over 1+ \cos x}.\]

Answer 1

(tan(x/2))^2

Answer 2

How?

Answer 3

things you need to know

\[\cos \theta = (1 - t^2)/1+t^2)\] then the problem is

\[[1 - (1 - t^2)/(1 + t^2)]/[1 +(1 - t^2)/(1 + t^2)]\]

let \[1 = (1+ t^2)/(1+t^2)\] this will allow for common denominators in the numerator and denominator

then \[[( 1+t^2 - 1+t^2)/(1+t^2)]/[( 1 +t^2 + 1 - t^2)/(1 + t^2)]\]

this simplifies to

\[[2t^2/(1+t^2)]/[2/(1+t^2)]\]

dividing by a fraction, flip the denominator and multiply

\[2t^2/(1+t^2) \times (1 + t^2)/2 = 2t^2/2\]

the answer is t^2 and

\[t = \tan(\theta/2)\]

so \[t^2 = (\tan(\theta/2))^2\]

= \[\tan^2(\theta/2)\]

Answer 4

Hmm. This doesn't look to simple... Is there an easier way?

Answer 5

nope thats it...I wrote all the steps... if you use t = tan(x/2) then cosx = (1 - t^2)/(1+t^2)

and sin x = 2t/(1+t^2)

its the most painful way to solve trig equations... sometimes called the method of last resort

Answer 6

Thanks anyway :)

Answer 7

if you know sin & cos... its just algebraic manipulation

Answer 8

hey @order check this method if this is an easy one or not
1-cos x=2sin^2(x/2)
1+cos x=2cos^2(x/2)
Substitute them and cancel the two to get tan^2(x/2). if you need these derivations i am ready to help

Answer 9

I can't seem to get this....

Answer 10

you know that
cosx=cos^2 (x/2)-sin^2 (x/2)
therefore cosx=cos^2 (x/2)-1+cos^2 (x/2) (cos^2 (x/2)=1- sin^2 (x/2))
therefore !+cos x=2cos^2 (x/2)
got it
similarly derive 1-cosx

Answer 11

then it shud be easy for you in 1-cosx convert the cos terms to sin terms

Answer 12

I'm sorry... My brain isn't functioning well today.. I'm completely lost! :(

Answer 13

where are you facing problem?

Answer 14

From everywhere! From the beginning....

Answer 15

hpoe you are aware of the formula \[\cos 2x=\cos ^{2}x-\sin^{2} x\]

Answer 16

from there you derive (1-cosx)
how?
cos2x=1-sin^2 x-sin^2 x
cos2x=1-2sin^x
1-cos2x=2sin^2 x

Answer 17

if 1-cos 2x=2sin^2 x
then 1-cosx=2 sin^2(x/2)