Average Power of load conected to an Ideal Transformer
This is, hopefully, just enough of an answer to get you to figure it out yourself.


Your voltage is given as a cosine function.
What is its peak voltage?
What is its RMS voltage?
What is the RMS current?
What happens when you ...?




As you have realised (indicated in the comments) in the question we are using \$ I_L \$ which is the peak value of a cosine waveform. The RMS value (required for the power calculation) is \$ \frac {1}{\sqrt 2} I_L \$.

Since \$ P = I^2R \$ we get \$ P_L = \left (\frac {1}{\sqrt 2} I_{pk} \right )^2 R_L = \frac {1}{2} I_{pk}^2 R_L  \$ as written in the question.