1. The given alternating current is $I(t)=I_A\,\sin \omega t+I_B\,\cos \omega t.$

2. The rms value is defined as $I_{\text{rms}}=\sqrt{\langle I^2(t)\rangle}.$ So first, square the current: $I^2=(I_A\sin \omega t+I_B\cos \omega t)^2.$

3. Expanding, $I^2=I_A^2\sin^2\omega t+I_B^2\cos^2\omega t+2I_AI_B\sin\omega t\cos\omega t.$

4. Now take average over one complete cycle: $\langle \sin^2\omega t\rangle=\frac12, \qquad \langle \cos^2\omega t\rangle=\frac12, \qquad \langle \sin\omega t\cos\omega t\rangle=0.$

5. Therefore, $\langle I^2\rangle=I_A^2\cdot \frac12+I_B^2\cdot \frac12+2I_AI_B\cdot 0$ $\Rightarrow \langle I^2\rangle=\frac{I_A^2+I_B^2}{2}.$

6. Hence the rms current is $I_{\text{rms}}=\sqrt{\frac{I_A^2+I_B^2}{2}}.$

7. Comparing with the options:

   • A: incorrect
   • B: correct
   • C: incorrect
   • D: incorrect

Therefore, the correct answer is Option B.