# Question 246a9

Mar 20, 2017

see below

#### Explanation:

Left Hand Side:

color(red)(1 + tan^2theta/(sec theta+1)=color(blue)(1+(sin^2theta/cos^2theta)/(1/costheta+1)

color(blue)(=1+(sin^2theta/cos^2theta)/((1+costheta)/cos theta)

color(blue)(=1+sin^2theta/cos^2theta*(cos theta)/(1+costheta)

color(blue)(=1+sin^2theta/cos^cancel2theta*cancel(cos theta)/(1+costheta)

color(blue)(=1+sin^2theta/(costheta(1+costheta)

color(blue)(=(cos theta+cos^2 theta+sin^2theta)/(costheta(1+cos theta))

color(blue)(=(cos theta+1)/(cos theta(1+cos theta)

color(blue)(=(1+cos theta)/(cos theta(1+cos theta)

color(blue)(=cancel(1+cos theta)/(cos theta cancel((1+cos theta)

color(blue)(=1/cos theta

color(blue)( :. = sec theta#

Mar 21, 2017

$L H S = 1 + {\tan}^{2} \frac{\theta}{\sec \theta + 1}$

$= 1 + \frac{{\sec}^{2} \theta - 1}{\sec \theta + 1}$

$= 1 + \frac{\left(\cancel{\sec \theta + 1}\right) \left(\sec \theta - 1\right)}{\cancel{\sec \theta + 1}}$

$= 1 + \sec \theta - 1 = \sec \theta = R H S$