\sin t=\frac{\sin t}{1}=\frac{\sin t}{\sqrt{\sin^2t+\cos^2t}}=\frac{{\sin t\over\cos t}}{\sqrt{{\sin^2t\over\cos^2t}+{\cos^2t\over\cos^2t}}}=\frac{\mathrm{tg}\,t}{\sqrt{\mathrm{tg}^2\,t+1}}