Question

What is \[\dfrac{{\tan C}}{{sin C}}\] equal to?

Answer 1

Hint: Here the question is related to trigonometry. The two terms which are present in the numerator and denominator are the trigonometric ratios. By using the definition of the trigonometric ratio and reciprocal relations of the trigonometric ratios we are determining the value of \[\dfrac{{\tan C}}{{sin C}}\].

Complete step by step answer:
The sine, cosine, tangent, cosecant, secant and cotangent are the trigonometry ratios of trigonometry. It is abbreviated as sin, cos, tan, cosec, sec and cot. We have totally 6 trigonometric ratios in trigonometry. The 4 trigonometry ratios are dependent on the other 2 trigonometric ratios. The trigonometric ratios are sine, cosine, tangent, cosecant, secant and cotangent. The tangent trigonometric ratio is the ratio of the sine and cosine trigonometric ratio.
\[ \Rightarrow \tan A = \dfrac{{\sin A}}{{\cos A}}\]

The cosecant trigonometric ratio is the reciprocal of the sine trigonometric ratio.
\[ \Rightarrow \csc A = \dfrac{1}{{\sin A}}\]
The secant trigonometric ratio is the reciprocal of the cosine trigonometric ratios.
\[ \Rightarrow \sec A = \dfrac{1}{{\cos A}}\]
The cotangent trigonometric ratio is the reciprocal of the tangent trigonometric ratio.
\[ \Rightarrow \cot A = \dfrac{1}{{\tan A}}\]
Consider the given question
\[\dfrac{{\tan C}}{{sin C}}\]
The C is the representation of the angle.
This can be written in the form of a product of two terms. i.e.,
\[ \Rightarrow \dfrac{{\operatorname{tan} \,C}}{{\operatorname{sin} \,C}} = \operatorname{tan} C.\dfrac{1}{{\operatorname{sin} C}}\]

As we know that the tangent trigonometric ratio is written as the ratio of the sine and cosine trigonometric ratio. So we have
\[ \Rightarrow \dfrac{{\operatorname{tan} \,C}}{{\operatorname{sin} \,C}} = \dfrac{{\operatorname{sin} C}}{{\operatorname{Cos} C}}.\dfrac{1}{{\operatorname{sin} C}} \\ \]
The \[\operatorname{sin} C\] gets canceled in both the numerator and denominator. So we have
\[ \Rightarrow \dfrac{{\operatorname{tan} \,C}}{{\operatorname{sin} \,C}} = \dfrac{1}{{\operatorname{Cos} C}} \\ \]
As we know, the secant trigonometric ratio is the reciprocal of the cosine trigonometric ratio. So the above inequality is written as
\[ \Rightarrow \dfrac{{\operatorname{tan} \,C}}{{\operatorname{sin} \,C}} = \operatorname{Sec} C\]

Therefore \[\dfrac{{\operatorname{tan} \,C}}{{\operatorname{sin} \,C}} = \operatorname{Sec} C\].

Note: The trigonometric ratios are not case sensitive, the ratios can be written in the form of capital also and in small letters also. Here the starting letter of the ratio is capital letter but the definition remains the same. Suppose a number or term is in the form of the division or in the fraction form it can be easily written down in the form product or multiplication form.