tan 30 * tan 60
-
0
-
1
-
sin 90
-
cos 0
-
tan 45
-
infinity
tan 30 = 1/sqrt(3) and tan 60 = sqrt(3). Their product is 1. sin 90, cos 0, and tan 45 are all equal to 1. Therefore, all marked options are mathematically correct.
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.