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I prove an identity between the first kind and the third kind completeelliptic integrals with the following form:  $$\Pi({(1+x) (1-3 x)\over (1-x) (1+3 x)}, {(1+x)^3(1-3 x)\over (1-x)^3(1+3x)})- {1+ 3 x \over 6 x} K ({(1+x)^3(1-3x)\over (1-x)^3 (1+3x)}) = 0, (0< x< 1); =-{\pi\over 12} {(x-1)^{3/2}\sqrt{1+3 x}\over x} (x<0 or x>1).$$  This relation can be applied to eliminate the complete elliptic integral ofthe third kind from the analytic solutions of the imaginary part of two-loopsunset diagrams in the equal mass case. The validity of this relation in thecomplex domain is also briefly discussed.

One Special Identity between the complete elliptic integrals of the first and the third kind