If $tan~A=Cot~B$, then $A+B=?$

Question

Accepted Answer

The correct option is 2:
To solve this problem, we need to recall the complementary angle identities in trigonometry. One of the key identities states that the cotangent of an angle is equal to the tangent of its complement. Mathematically, this can be expressed as:
$$Cot\~B = tan\~(90^{\circ} - B)$$
We are given the equation:
$$tan\~A = Cot\~B$$
Now, substitute the complementary angle identity for $Cot\~B$ into the given equation:
$$tan\~A = tan\~(90^{\circ} - B)$$
Since the tangent function is equal for both angles, we can equate the angles (assuming A and B are acute angles, which is typical in such problems unless otherwise specified):
$$A = 90^{\circ} - B$$
Now, rearrange the equation to find the sum of A and B:
$$A + B = 90^{\circ}$$
Therefore, if $tan\~A = Cot\~B$, then $A+B = 90^{\circ}$.

The incorrect option is 1: $45^{\circ}$. This would imply that $A+B$ is half of $90^{\circ}$. This is incorrect because the complementary angle identity directly leads to a sum of $90^{\circ}$.
The incorrect option is 3: $180^{\circ}$. This value would typically be associated with supplementary angles or certain trigonometric identities involving $\pi$ radians, not with the relationship between tangent and cotangent of complementary angles.
The incorrect option is 4: $360^{\circ}$. This value represents a full rotation and is not related to the complementary angle relationship between tangent and cotangent.