To solve the given expression, we can use the trigonometric identity:
[ \tan(A) \cdot \tan(B) = \frac{{\sin(A) \cdot \sin(B)}}{{\cos(A) \cdot \cos(B)}} ]
Let's evaluate each option to determine if it is correct or not:
Option A) 0 - This option is incorrect because the product of the tangents of two angles is not necessarily zero.
Option B) 1 - This option is correct. By substituting the values of the angles, we have:
[ \tan(30) \cdot \tan(60) = \frac{{\sin(30) \cdot \sin(60)}}{{\cos(30) \cdot \cos(60)}} = \frac{{\frac{1}{2} \cdot \frac{\sqrt{3}}{2}}}{{\frac{\sqrt{3}}{2} \cdot \frac{1}{2}}} = 1 ]
Option C) sin 90 - This option is correct. By substituting the values of the angles, we have:
[ \tan(30) \cdot \tan(60) = \frac{{\sin(30) \cdot \sin(60)}}{{\cos(30) \cdot \cos(60)}} = \frac{{\frac{1}{2} \cdot \frac{\sqrt{3}}{2}}}{{\frac{\sqrt{3}}{2} \cdot \frac{1}{2}}} = 1 ]
Since (\sin(90) = 1), this option is correct.
Option D) cos 0 - This option is correct. By substituting the values of the angles, we have:
[ \tan(30) \cdot \tan(60) = \frac{{\sin(30) \cdot \sin(60)}}{{\cos(30) \cdot \cos(60)}} = \frac{{\frac{1}{2} \cdot \frac{\sqrt{3}}{2}}}{{\frac{\sqrt{3}}{2} \cdot \frac{1}{2}}} = 1 ]
Since (\cos(0) = 1), this option is correct.
Option E) tan 45 - This option is correct. By substituting the values of the angles, we have:
[ \tan(30) \cdot \tan(60) = \frac{{\sin(30) \cdot \sin(60)}}{{\cos(30) \cdot \cos(60)}} = \frac{{\frac{1}{2} \cdot \frac{\sqrt{3}}{2}}}{{\frac{\sqrt{3}}{2} \cdot \frac{1}{2}}} = 1 ]
Since (\tan(45) = 1), this option is correct.
Therefore, the correct options are B, C, D, and E.