Question

$\frac{{sin}^4\theta -{cos}^4\theta +1}{{cos}^2\theta };(\theta \ne {90}^{\circ })$ equals

• A. $2ta{n}^2\theta$
• B. $2$
• C. $2se{c}^7\theta$
• D. $1$

Answer 1

The correct option is A $2ta{n}^2\theta$

$\frac{{\sin }^4\theta -{\cos }^4\theta +1}{{\cos }^2\theta }$

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

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

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

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

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

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