# help on proving trig identity

• Nov 22nd 2009, 08:41 AM
help on proving trig identity
prove the following identity:

Secθ- 1/ Secθ +1 ≡ Tan² θ/2 (only the theta is divided by 2)

this is what i've done so far:

LHS =( secθ- 1)/ (secθ +1)

= (1/cosθ - 1) /(1/cosθ +1 )

= 1- cosθ/ 1+ cosθ

from here, I don't know what to do next. help plz!!!
• Nov 22nd 2009, 08:52 AM
Quote:

prove the following identity:

Secθ- 1/ Secθ +1 ≡ Tan² θ/2 (only the theta is divided by 2)

this is what i've done so far:

LHS =( secθ- 1)/ (secθ +1)

= (1/cosθ - 1) /(1/cosθ +1 )

= 1- cosθ/ 1+ cosθ

from here, I don't know what to do next. help plz!!!

You're doing great so far!

Now use
$\displaystyle \cos\theta = 1-2\sin^2(\tfrac12\theta)$
on the top, and
$\displaystyle \cos\theta = 2 \cos^2(\tfrac12\theta)-1$
on the bottom, and you're there!

• Nov 22nd 2009, 09:05 AM
Quote:

Hello ladythugetteYou're doing great so far!

Now use
$\displaystyle \cos\theta = 1-2\sin^2(\tfrac12\theta)$
on the top, and
$\displaystyle \cos\theta = 2 \cos^2(\tfrac12\theta)-1$
on the bottom, and you're there!

$\displaystyle cos(\theta) = cos\left(\frac{\theta}{2} + \frac{\theta}{2}\right) = cos \left(2\, \frac{\theta}{2}\right)$