5. Докажите тождество: \( \frac{tgt}{tgt + ctgt} = \sin^2 t \).

\( \frac{tgt}{tgt + ctgt} = \frac{\frac{\sin t}{\cos t}}{\frac{\sin t}{\cos t} + \frac{\cos t}{\sin t}} = \frac{\frac{\sin t}{\cos t}}{\frac{\sin^2 t + \cos^2 t}{\sin t \cos t}} = \frac{\sin t}{\cos t} \cdot \frac{\sin t \cos t}{\sin^2 t + \cos^2 t} = \frac{\sin t}{\cos t} \cdot \frac{\sin t \cos t}{1} = \sin^2 t \) Таким образом, \( \frac{tgt}{tgt + ctgt} = \sin^2 t \).