Why phasor multiplication don't agree with sinusoid multiplication?

let us we have two sinusoids sin(wt+30) and sin(wt+45). if we multiply them, when t=0s, we get sin30xsin45=.35355339 Now if we transform these two sinusoids into phasor, we get (co30+j sin30) and (cos45+j sin45) respectively. Multiplying these two phasors we get a new phasor (cos75+j sin75). This new phasor should represent the result sinusoid of above multiplication of two sinusoid, as in phasor addition. But at t=0s, this phasor shows sin75. But clearly sin30xsin45 is not equal to sin75. So why do we use this in AC circuit analysis? I know I am doing something wrong, but where is it?

#Algebra

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Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
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