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 ...?

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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.