How to proceed step by step while taking derivatives of inverse trigonometric functions?

While trying to find a derivative of the inverse trignometric functions, apply the chain rule that you normally apply to the other functions also. 

So, first compute the derivate of the inverse, then find the derivative of the function thats inside inverse function. 

 

For example if  f(x) = sin^-1 (x^2) and you need to find the derivate of f(x)

f'(x) = derivative of (sin^-1 (x^2)) * derivate of (x^2)

f'(x) = 1/(sqrt(1-(x^2)^2)) * 2x

f'(x) = 1/sqrt(1-x^4)* 2x

f'(x) = 2x/sqrt(1-x^4) 

 

In response to your second question, there is no defined way of determining the use of a trignometric identity. You should remember all the trignometric identities by heart, so that whenever you will se a question which has any of the terms, you can immediately use them. Also, practice as much as you can, so that you reach a point where just by looking at the question, you can determine which identity to use.