You are a computer programmer working for a company that provides online educational resources to their consumers. Your boss has tasked you and a group of other programmers to create a user friendly mobile app that is capable of taking a math equation, simplifying it to its simplest form, and showing the user a step$-$by$-$step solution of how the equation was simplified. Your boss believes that this app will be just the resource the company needs to bring better quality math material to their consumers. You have the responsibility in the group to program the trigonometry section on the app.

Not only do you want the app to simplify the expression, you want the app to tell the user whether or not the trig equation simplifies to a trig identity or not. You have been given the following statements as test cases:

Equation $1$:

Equation $2$:

Equation $3$:

Do the following for each equation:

a$.$ Verify if each equation is a trig identity.

b$.$ Explain why or why not.

c$.$ How beneficial do you think it would be to you, the programmer, to have simplified equations even if you could just copy and paste all the relevant code and still get the same end result? Explain your reasonings either for or against.