[tex]a. A=\frac{1+2sin\alpha .cos\alpha }{cos^{2}\alpha -sin^{2}\alpha }[/tex]
[tex]b. B=(1+tan^{2}\alpha ).(1-sin^{2}\alpha )-(1+cot^{2}\alpha ).(1-cos^{2}\alpha )[/tex]
$a)A=\dfrac{1+2sin\alpha .cos\alpha }{cos^{2}\alpha -sin^{2}\alpha }=\dfrac{\sin^2 \alpha +\cos^2 \alpha +2\sin \alpha .\cos \alpha }{(\cos \alpha -\sin \alpha )(\cos \alpha +\sin \alpha )}
\\=\dfrac{(\cos \alpha +\sin \alpha )^2}{(\cos \alpha -\sin \alpha )(\cos \alpha +\sin \alpha )}=\dfrac{\cos \alpha +\sin \alpha }{\cos \alpha -\sin \alpha }
\\b)B=(1+\tan^2 \alpha )(1-\sin^2\alpha )-(1+\cot^2 \alpha )(1-\cos^2 \alpha)
\\=\dfrac{1}{\cos^2 \alpha }.\cos^2 \alpha -\dfrac{1}{\sin^2 \alpha }.\sin^2 \alpha
\\=1-1=0$