Show that for a wave travelling on a string

where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to write

?

Show that the particle speed can never be equal to the wave speed in a sine wave if the amplitude is less than wavelength divided by 2π.

Two wave pulses identical in shape but inverted with respect to each other are produced at the two ends of a stretched string. At an instant when the pulses reach the middle, the string becomes completely straight. What happens to the energy of the two pulses?

What is the smallest positive phase constant which is equivalent to 75 π?

A string clamped at both ends vibrates in its fundamental mode. Is there any positive (except the ends) on the string which can be touched without disturbing the motion? What if the string vibrates in its first overtone?