Using an inductive argument to show that

My attempts so far are:

Step 1. See if it's true for n=1. So,

So LHS = RHS (I think I can say this based on the identity sin2A=2sinAcosA? This is my first issue)

Step 2. Assume it holds true for n = k

So,

Step 3. Show it's true for k+1

Here I know if it is true, then

So taking the LHS

Using index laws

Using sin2A = 2sinAcosA

Now here is where I am confused. I know back in Step 2 I have and need to substitute this in but I'm unsure how to resolve it to get the answer I know I need!

Hope this all makes sense and that someone can give me a hand. Thanks very much.