Question

If ${\sin }^4\theta +{\sin }^2\theta =1$ then prove that ${\tan }^4\theta -{\tan }^2\theta =1$.

Answer 1

Given,

${\sin }^4\theta +{\sin }^2\theta =1$

or, ${\sin }^4\theta =1-{\sin }^2\theta$

or, ${\sin }^4\theta ={\cos }^2\theta$.......(1).

Now,

${\tan }^4\theta -{\tan }^2\theta$

$=\frac{{\sin }^4\theta }{{\cos }^4\theta }-{\tan }^2\theta$

$=\frac{1}{{\cos }^2\theta }-{\tan }^2\theta$[ Using (1)]

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

$=1$.