Getting value of $\sin A \sin B$

Question

my solution

I am able to get the answer. When i am trying to evaluate Sin A Sin B , i am not clear how to approach? I feel data is insufficeint to discredit option D

Answer 1

Since $$\tan A\tan B = 2\implies \sin A \sin B= 2\cos A\cos B$$ and $$ \cos(A-B) = \cos A\cos B+\sin A \sin B ={3\over 5}$$

we get $$\cos A\cos B ={1\over 5}\implies \sin A \sin B = {2\over 5}$$

Answer 2

Given:

$$C_aC_b+S_aS_b=\frac35,$$ $$S_aS_b=2\,CaCb.$$

a) $3\,C_aC_b=\dfrac35$;

b) $S_aS_b=2\,CaCb$;

c) $\cos(a+b)=C_aC_b-S_aS_b=\dfrac15-\dfrac25$;

d) $\sin(a-b)=\pm\sqrt{1-\cos^2(a-b)}=\pm\dfrac45, \\\sin(a+b)=\pm\sqrt{1-\cos^2(a+b)}=\pm\dfrac{\sqrt{24}}5, \\2S_aC_b=\sin(a-b)+\sin(a+b)=\dfrac{\pm4\pm\sqrt{24}}5.$