Question

Can someone help me solve this 11

Answer 1

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

`=-\left(\frac{1}{\tfrac{1- \tan^2 \alpha}{1 + \tan^2 \alpha}} \right)~~~~~~~~~~~~~~~~~~; \left[\cos 2\theta = (1 - \tan^2 \theta)/(1+ \tan^2 \theta) \right]\\\\\\\ = -\left((1+ \tan^2 \alpha)/(1- \tan^2 \alpha)\right)\\\\\\= (1+\tan^2 \alpha)/(\tan^2 \alpha -1)`