Question

How do you simplify \[\dfrac{{\cos x}}{{\sin x}}\]?

Answer 1

Hint: Trigonometric functions are defined as real functions which relate any angle of a right angled triangle to the ratio of any two of its sides. We may use geometric definitions to evaluate trigonometric values. Here, it’s important that we know the cosine of theta is the ratio of the adjacent side (base) to the hypotenuse and the sine of theta is the ratio of the opposite side to the hypotenuse.

Complete step-by-step answer:
According to the given data, we need to simplify \[\dfrac{{\cos x}}{{\sin x}}\]
If in a right angled triangle \[\theta \] represents one of its acute angle then by definition we can write
\[\cos \theta = \dfrac{{Base}}{{Hypotenuse}}\] and,
\[\sin \theta = \dfrac{{Opposite}}{{Hypotenuse}}\]
Here we need to evaluate,
\[\dfrac{{\cos \theta }}{{\sin \theta }} = \cos \theta \times \dfrac{1}{{\sin \theta }} = \dfrac{{Base}}{{Hypotenuse}} \times \dfrac{{Hypotenuse}}{{Opposite}}\]
Hence, we finally get,
\[\dfrac{{\cos \theta }}{{\sin \theta }} = \dfrac{{Base}}{{Opposite}}\]
Also, we know that the cotangent of theta is the ratio of the adjacent side (base) to the opposite side.
According to the given data, \[\theta = x\].
Hence, when we substitute the value in the expression, we get
\[ \Rightarrow \dfrac{{\cos x}}{{\sin x}} = \dfrac{{Base}}{{Opposite}} = \cot x\]

Hence, the value of \[\dfrac{{\cos x}}{{\sin x}}\] is equivalent to \[\cot x\].

Note: Trigonometric functions are real functions which relate any angle of a right angled triangle to the ratio of any two of its sides. One should be careful while evaluating trigonometric values and rearranging the terms to convert from one function to the other. The widely used ones are sin, cos and tan. While the rest can be referred to as the inverse of the other trigonometric ratios, i.e., cosec, sec and cot respectively. If in a right angled triangle θ represents one of its acute angles then, \[\cot x = \dfrac{{\cos x}}{{\sin x}}\].