An AC current is given by $$I = I_1 \sin\omega t + I_2 \cos\omega t$$. A hot wire ammeter will give a reading:

The given current is $$I = I_1 \sin\omega t + I_2 \cos\omega t$$. A hot wire ammeter reads the RMS (root mean square) value of the current.

The RMS value is $$I_{\text{rms}} = \sqrt{\langle I^2 \rangle}$$, where $$\langle \cdot \rangle$$ denotes the time average. Expanding $$I^2 = I_1^2 \sin^2\omega t + I_2^2 \cos^2\omega t + 2I_1 I_2 \sin\omega t \cos\omega t$$.

Taking the time average: $$\langle \sin^2\omega t \rangle = \frac{1}{2}$$, $$\langle \cos^2\omega t \rangle = \frac{1}{2}$$, and $$\langle \sin\omega t \cos\omega t \rangle = 0$$. Therefore $$\langle I^2 \rangle = \frac{I_1^2}{2} + \frac{I_2^2}{2} = \frac{I_1^2 + I_2^2}{2}$$.

Thus $$I_{\text{rms}} = \sqrt{\frac{I_1^2 + I_2^2}{2}}$$, and the correct answer is option 2.