Question

Prove that: $(1-\cos 2\theta ){\sec }^2\theta =2{\tan }^2\theta$

Answer 1

$(1-\cos 2\theta ){\sec }^2\theta =2{\tan }^2\theta$

L.H.S.$(1-\cos 2\theta ){\sec }^2\theta$

$=(1-(1-2{\sin }^2\theta \left)\right){\sec }^2\theta$

$=(1-1+2{\sin }^2\theta ){\sec }^2\theta$

$=2{\sin }^2\theta {\sec }^2\theta$

$=\frac{2{\sin }^2\theta }{{\cos }^2\theta }$

$=2{\tan }^2\theta$

$=$ R.H.S.