# Question #b18ed

Feb 15, 2017

2

#### Explanation:

Develop the numerator, after replacing 1 by (sin^2 x + cos^2 x) -->
$N = {\sin}^{2} x + {\cos}^{2} x - {\sin}^{4} x - {\cos}^{4} x =$
$= {\sin}^{2} x \left(1 - {\sin}^{2} x\right) + {\cos}^{2} x \left(1 - {\cos}^{2} x\right) =$
Use trig identities
$1 - {\sin}^{2} x = {\cos}^{2} x$
$1 - {\cos}^{2} x = {\sin}^{2} x$
$N = {\sin}^{2} x . {\cos}^{2} x + {\cos}^{2} x . {\sin}^{2} x$
Denominator:$D = {\sin}^{2} x . {\cos}^{2} x$
The expression becomes:
$E = \frac{N}{D} = 1 + 1 = 2$

Feb 15, 2017

$\frac{1 - \left({\sin}^{4} x + {\cos}^{4} x\right)}{{\cos}^{2} x {\sin}^{2} x}$

$= \frac{1 - \left({\left({\sin}^{2} x + {\cos}^{2} x\right)}^{2} - 2 {\sin}^{2} x {\cos}^{2} x\right)}{{\cos}^{2} x {\sin}^{2} x}$

$= \frac{1 - \left({1}^{2} - 2 {\sin}^{2} x {\cos}^{2} x\right)}{{\cos}^{2} x {\sin}^{2} x}$

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

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