What is the practical use of proving so many trigonometrical identities for school maths classess?
It is actually the most useful subject.
Rechargeable batteries we use in most of our electronics. When we use electricity to charge our batteries, we are converting alternating current to direct current. The alternating current has a shape of a sine curve f(x)=sinx and the direct current has a shape of f(x)=|sinx|
Suppose there are musical robots which produce musical notes beautifully. A team of engineers, designers and computer programmers are needed. In their course of work, they will need to alter the magnitude and frequency of the sound wave produced, many of which will involve equations such as y=a sin mx + b cos mx
Trigonometry is useful in many fields to explain the behavior of specific phenomena within. Some of the fields that uses trigonmetric identities are: architecture, civil engineering, mechanical engineering, and even music theory. For instance, in the music theory, the vibration of a string has the same shape as the sine function has. In other fields, such as biology or the study of climates, the periodic nature of sine and cosine functions helps to study and describe very well many of the distinctive processes.
To prove a basic trigonometric identity is the best way to learn it and to manage it in order to use it as a tool in solving more difficult trigonometric problems.