Question

Prove ${\sec }^4\theta (1-{\sin }^4\theta )-2{\tan }^2\theta =1$

Answer 1

L.H.S

${\sec }^4\theta (1-{\sin }^4\theta )-2{\tan }^2\theta$

${\sec }^4\theta ({\left(1^2\right)}^2-{\left({\sin }^2\theta \right)}^2)-2{\tan }^2\theta$

$={\sec }^4\theta (1+{\sin }^2\theta )(1-{\sin }^2\theta )-2{\tan }^2\theta$

$=\frac{1}{{\cos }^2\theta {\cos }^2\theta }(1+{\sin }^2\theta ){\cos }^2\theta -2{\tan }^2\theta$

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

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

$=\frac{1-{\sin }^2\theta }{{\cos }^2\theta }$

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

$=1$

R.H.S

Hence, proved.