# Question 08381

Mar 21, 2017

1

#### Explanation:

Remember $\cot \left(x\right) = \cos \frac{x}{\sin} \left(x\right)$

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

((cos^2(x)/sin^2(x))(cos^2(x)/1))/((cos^2(x)/sin^2(x))-((cos^2(x)sin^2(x))/sin^2(x))

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

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

cos^4(x)/(cos^2(x)(1-sin^2(x))#

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

Use trig identities, specifically:
${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$
${\cos}^{2} \left(x\right) = 1 - {\sin}^{2} \left(x\right)$

${\cos}^{2} \frac{x}{\cos} ^ 2 \left(x\right)$

1

Mar 27, 2017

$\frac{{\cot}^{2} x {\cos}^{2} x}{{\cot}^{2} x - {\cos}^{2} x}$

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

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

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

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