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In a #DeltaABC#,
#rarrcosA=(b^2+c^2-a^2)/(2bc)#
#rarrArea=(1/2)a*bsinC#
Now, #Area# of #DeltaABD=(1/2)*9*8*sinx=36sinx#
#Area# of #DeltaADC=(1/2)*8*18*sinx=72sinx#
#Area# of #DeltaABC=(1/2)*9*18*sin2x=81sin2x#
#rarrDeltaABC=DeltaABD+DeltaADC#
#rarr81sin2x=36*sinx+72*sinx=108*sinx#
#rarr81*2cancel(sinx)*cosx=108*cancel(sinx)#
#rarrcosx=(108)/162=2/3#
Applying cosine law in #DeltaABC#, we get,
#rarrcos2x=(9^2+18^2-a^2)/(2*9*18)#
#rarr2cos^2x-1=(405-a^2)/324#
#rarr2*(2/3)^2-1=(405-a^2)/324#
#rarr2*(4/9)-1=(405-a^2)/324#
#rarr-36=405-a^2#
#rarra^2=405+36=441#
#rarra=21#
Also, note that
#rarrsin2x=2sinxcosx#
#rarrcos2x=2cos^2x-1#
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