Question

Find the value of \[\tan {30^ \circ }\cot {30^ \circ }\].

Answer 1

Hint:
In trigonometry the terms tan and cot stand for tangent and cotangent respectively. We have some formulas in trigonometry which relate these two ratios. One of them is \[\tan \theta = \dfrac{1}{{\cot \theta }}\]. Using this relation the given problem can be evaluated.

Complete step by step solution:
Tan or cot functions stand for tangent and cotangent of an angle respectively in trigonometry. The tangent of an angle is also defined by the ratio of the sin of the angle to the cosine of the angle, while the cotangent is the ratio of the cosine of the angle to the sine of the angle.
Now the cotangent of an angle is the reciprocal of the tangent of the angle.
We have to evaluate \[\tan {30^ \circ }\cot {30^ \circ }\]….(1)
Now we can rewrite \[\tan {30^ \circ } = \dfrac{1}{{\cot {{30}^ \circ }}}\]…(2)
Substituting (2) in (1) we get:
\[\tan {30^ \circ }\cot {30^ \circ }\] \[ = \dfrac{1}{{\cot {{30}^ \circ }}}\cot {30^ \circ }\]
\[ \Rightarrow \] \[\tan {30^ \circ }\cot {30^ \circ }\] \[ = 1\]
Observe that we could also rewrite \[\cot {30^ \circ } = \dfrac{1}{{\tan {{30}^ \circ }}}\] and substitute in equation (1), to get the same result.

Hence, \[\tan {30^ \circ }\cot {30^ \circ }\] \[ = 1\].

Note:
This problem could also be evaluated by using the values of the standard angles. Students must memorise the values of \[{0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ },{90^ \circ }\] of the trigonometric functions \[\sin ,\cos ,\tan ,\cot ,\sec ,\cos ec\].
Now \[\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}\]
\[\cot {30^ \circ } = \sqrt 3 \]
\[\therefore \] \[\tan {30^ \circ }\cot {30^ \circ }\] \[ = \dfrac{1}{{\sqrt 3 }}\sqrt 3 \] \[ = 1\].
Note that none of the trigonometric ratios have any unit as they are ratios of similar quantities.