Question

Prove that

$(sec\theta +cos\theta )(sec\theta -cos\theta )={tan}^2\theta +{sin}^2\theta$

Answer 1

$L.H.S=(\sec \theta +cos\theta )(\sec \theta -\cos \theta )$

$={\sec }^2\theta -{\cos }^2\theta$

$=1+{\tan }^2\theta -{\cos }^2\theta [\because 1+{\tan }^2\theta ={\sec }^2\theta ]$

$={\tan }^2\theta +1-{\cos }^2\theta$

$={\tan }^2\theta +{\sin }^2\theta [\because {\sin }^2\theta +{\cos }^2\theta =1]$

$=R.H.S$

$\therefore (\sec \theta +cos\theta )(\sec \theta -\cos \theta )={\tan }^2\theta +{\sin }^2\theta$