[tex]\dfrac{(\sin \alpha+\cos \alpha)^2-1}{2\tan \alpha}=\\\\\dfrac{\sin^2 \alpha+2\sin \alpha\cos \alpha+\cos^2 \alpha-1}{\dfrac{2\sin \alpha}{\cos \alpha}}=\\\\(1+2\sin \alpha\cos \alpha-1)\cdot \dfrac{\cos \alpha}{2\sin \alpha}}=\\\\2\sin \alpha\cos \alpha\cdot\dfrac{\cos \alpha}{2\sin \alpha}}=\\\\\cos^2 \alpha[/tex]
