From the top of a lighthouse, 100 m high, the angle of depression of a boat is \( \tan^{-1}\left(\frac{5}{12}\right) \). What is the distance between the boat and the lighthouse?
The maximum value of \( \sin\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right) \) in the interval \( \left(0, \frac{\pi}{2}\right) \) is attained at
Let x, y, z be positive real numbers such that x, y, z are in GP and \( \tan^{-1}x \), \( \tan^{-1}y \) and \( \tan^{-1}z \) are in AP. Then which one of the following is correct?
Consider the following for triangle ABC:
1. \( \sin\left(\frac{B+C}{2}\right) = \cos\left(\frac{A}{2}\right) \)
2. \( \tan\left(\frac{B+C}{2}\right) = \cot\left(\frac{A}{2}\right) \)
3. \( \sin(B+C) = \cos A \)
4. \( \tan(B+C) = -\cot A \)
Which of the above are correct?
