Question

Prove the following identity:
\[\frac{ \sin \theta -\tan \theta }{ \sin \theta +\tan \theta }=\frac{ 1-\sec \theta }{ 1+\sec }\]

Answer 1

write tan t = sin t/ cos t
then take out sin t common from numerator and denominator, and cancel them.
whats left ?

Answer 2

did u understand that @Beatles ?

Answer 3

cos t sin t - sin t / cos t all over cos t sin t + sin t / cos t
it will back from the start.

Answer 4

huh ?
just take sin x common from numerator
what u get ?

Answer 5

that's what I get.

Answer 6

cos t sin t - sin t = sin t (cos t-1)
ok ?

Answer 7

sin t (cos t - 1) /cos t

Answer 8

yup

Answer 9

and denominator is sin t (cos t +1)right ?

Answer 10

yup.

Answer 11

then what's next?

Answer 12

so cancel those sin x in numerator and denominator
whats left is only
cos t-1 / cos t+1

Answer 13

now write cos t = 1/ sec t

Answer 14

ok?

Answer 15

no!

Answer 16

if you have sin t (cos t - 1) they have a dinaminator of cos t.

Answer 17

if you get the identity of tan t = sin t/ cos t right? so sin t - sin t/cos t

Answer 18

that cos t in denominator was cancelled with cos t in denominator of (sin t+ tan t)

Answer 19

okay I get it. :)

Answer 20

thanks. :)

Answer 21

welcome :)