# Simplify the expression:? (sin^2(pi/2+alpha)-cos^2(alpha-pi/2))/(tg^2(pi/2+alpha)-ctg^2(alpha-pi/2))

## $\frac{{\sin}^{2} \left(\frac{\pi}{2} + \alpha\right) - {\cos}^{2} \left(\alpha - \frac{\pi}{2}\right)}{t {g}^{2} \left(\frac{\pi}{2} + \alpha\right) - c t {g}^{2} \left(\alpha - \frac{\pi}{2}\right)}$

Apr 10, 2017

$\frac{{\sin}^{2} \left(\frac{\pi}{2} + \alpha\right) - {\cos}^{2} \left(\alpha - \frac{\pi}{2}\right)}{{\tan}^{2} \left(\frac{\pi}{2} + \alpha\right) - {\cot}^{2} \left(\alpha - \frac{\pi}{2}\right)}$

$= \frac{{\sin}^{2} \left(\frac{\pi}{2} + \alpha\right) - {\cos}^{2} \left(\frac{\pi}{2} - \alpha\right)}{{\tan}^{2} \left(\frac{\pi}{2} + \alpha\right) - {\cot}^{2} \left(\frac{\pi}{2} - \alpha\right)}$

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

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

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

$= \frac{{\cos}^{2} \left(\alpha\right) - {\sin}^{2} \left(\alpha\right)}{{\cos}^{4} \left(\alpha\right) - {\sin}^{4} \left(\alpha\right)} \times \frac{{\sin}^{2} \left(\alpha\right) {\cos}^{2} \left(\alpha\right)}{1}$

=(cos^2(alpha)-sin^2(alpha))/((cos^2(alpha)-sin^2(alpha))(cos^2(alpha)+sin^2(alpha))xx(sin^2(alpha)cos^2(alpha))/1

$= {\sin}^{2} \left(\alpha\right) {\cos}^{2} \left(\alpha\right)$