Hi All,
This reads as
one over sec theta + tan theta = sec theta - tan theta = cos theta over one + sin theta
1 / secѳ + tanѳ = sec ѳ – tanѳ = cosѳ / 1 + sinѳ
My job is to prove the identities
Any help would be appreciated
Second Identity:
$\displaystyle \frac{1}{\sec{\theta} + \tan{\theta}} = \frac{1}{\frac{1}{\cos{\theta}} + \frac{\sin{\theta}}{\cos{\theta}}}$
$\displaystyle = \frac{1}{\frac{1 + \sin{\theta}}{\cos{\theta}}}$
$\displaystyle = \frac{\cos{\theta}}{1 + \sin{\theta}}$
First identity:
$\displaystyle = \frac{\cos{\theta}(1 - \sin{\theta})}{1 - \sin^2{\theta}}$
$\displaystyle = \frac{\cos{\theta} - \cos{\theta}\sin{\theta}}{\cos^2{\theta}}$
$\displaystyle = \frac{1}{\cos{\theta}} - \frac{\sin{\theta}}{\cos{\theta}}$
$\displaystyle = \sec{\theta} - \tan{\theta}$.