Why do you have to leave one side alone when proving a sentence to be true?

This question relates to identifying trigonometric identities to be true, however, my teacher says this question applies to everything. I do not understand this question, let alone, the answer. Can anyone help?

When proving a sentence to be true, particularly in the context of trigonometric identities, it is important to focus on one side of the equation and leave the other side alone. This is because when you manipulate an equation, such as adding, subtracting, multiplying, or dividing both sides by the same value, you are essentially performing an operation that changes the equation.

By leaving one side alone and only manipulating the other side, you can clearly demonstrate that both sides of the equation are equal. This approach allows you to break down the more complex side of the equation and simplify it step by step until it matches the other side.

Leaving one side unchanged serves as a comparison or reference point for the changes you make on the other side. If you manipulate both sides simultaneously, it can become difficult to determine whether the equality has been preserved or if any mistakes have been made along the way.

To illustrate this concept, let's consider an example: proving the trigonometric identity sin²θ + cos²θ = 1.

Start with the left side of the equation (sin²θ + cos²θ) and keep the right side (1) unchanged. Then, use trigonometric identities, algebraic manipulations, and other mathematical properties to transform the left side until it simplifies to match the right side. This may involve using Pythagorean identities, factoring, or rearranging terms.

By focusing solely on one side and leaving the other side untouched, you can methodically and logically show that both sides are equivalent, proving the desired trigonometric identity to be true.

Remember, this concept of leaving one side alone is not limited to trigonometric identities but can be applied to various mathematical proofs and equations across different topics. It helps maintain clarity and ensures the validity of your reasoning and steps throughout the proof.