Question

If \[5\cot \theta =12\], find the value of:
\[\operatorname{cosec}\theta +sec\theta \]

Answer 1

Hint: In this question, we first need to know about some of the basic definitions of trigonometry. Then expand the given equation by using the trigonometric identities to simplify it further.

Complete step-by-step answer:

\[\begin{align}
  & \cot \theta =\dfrac{1}{\tan \theta } \\
 & \sec \theta =\dfrac{1}{\cos \theta } \\
 & \operatorname{cosec}\theta =\dfrac{1}{\sin \theta } \\
\end{align}\]
Let us look at some of the basic definitions.
ANGLE: When a ray OA starting from its initial position OA rotates about its end point O and takes the final position OB, we say that angle AOB is formed.
The amount of rotation from initial side to the terminal side is called the measure of the angle.
POSITIVE AND NEGATIVE ANGLES:
An angle formed by a rotating ray is said to be positive or negative depending on whether it moves in an anti-clockwise or a clockwise direction respectively.
TRIGONOMETRIC RATIOS:
Relations between different sides and angles of a right angled triangle are called trigonometric ratios.
\[\begin{align}
  & \sin \theta =\dfrac{\text{Opposite}}{\text{Hypotenuse}} \\
 & \cos \theta =\dfrac{\text{adjacent}}{\text{Hypotenuse}} \\
 & \tan \theta =\dfrac{\text{Opposite}}{\text{adjacent}} \\
\end{align}\]
TRIGONOMETRIC IDENTITIES:
An equation involving trigonometric functions which is true for all those angles for which the functions are defined is called trigonometric identity.
\[\begin{align}
  & \cot \theta =\dfrac{1}{\tan \theta } \\
 & \sec \theta =\dfrac{1}{\cos \theta } \\
 & \operatorname{cosec}\theta =\dfrac{1}{\sin \theta } \\
\end{align}\]
Now, from the given equation in the question on rearranging the terms we have.
\[\begin{align}
  & \Rightarrow \cot \theta =\dfrac{12}{5} \\
 & \Rightarrow \tan \theta =\dfrac{5}{12}\text{ }\left[ \because \cot \theta =\dfrac{1}{\tan \theta } \right] \\
\end{align}\]
Now, let us convert the tangent form above to sine and cosine form using the corresponding trigonometric ratios formulae.
On comparing the tangent value we have to the tangent formula we get,
Adjacent side as 12 and opposite side as 5 in a right angle triangle.
Now, we can get the hypotenuse by using the formula,
\[\begin{align}
  & \Rightarrow \text{hypotenuse}=\sqrt{{{5}^{2}}+{{12}^{2}}} \\
 & \Rightarrow \text{hypotenuse}=\sqrt{169} \\
 & \therefore \text{hypotenuse}=13 \\
\end{align}\]
\[\left[ \begin{align}
  & \because \sin \theta =\dfrac{\text{Opposite}}{\text{Hypotenuse}} \\
 & \cos \theta =\dfrac{\text{adjacent}}{\text{Hypotenuse}} \\
 & \tan \theta =\dfrac{\text{Opposite}}{\text{adjacent}} \\
\end{align} \right]\]
By using the above formula we get,
\[\Rightarrow \sin \theta =\dfrac{5}{13}\]
\[\Rightarrow \cos \theta =\dfrac{12}{13}\]
Now, from the trigonometric identities we can get the relation between sine and cosecant and cosine and secant.
On substituting the corresponding values in the trigonometric identities we get,
\[\begin{align}
  & \Rightarrow \operatorname{cosec}\theta =\dfrac{13}{5}\text{ }\left[ \because \operatorname{cosec}\theta =\dfrac{1}{\sin \theta } \right] \\
 & \Rightarrow \sec \theta =\dfrac{13}{12}\text{ }\left[ \because \sec \theta =\dfrac{1}{\cos \theta } \right] \\
\end{align}\]
Now, on adding the values and further simplifying it we get,
\[\begin{align}
  & \Rightarrow \operatorname{cosec}\theta +\sec \theta \\
 & \Rightarrow \dfrac{13}{5}+\dfrac{13}{12} \\
\end{align}\]
\[\begin{align}
  & \Rightarrow 13\left( \dfrac{1}{5}+\dfrac{1}{12} \right) \\
 & \Rightarrow 13\left( \dfrac{12+5}{5\times 12} \right) \\
\end{align}\]
\[\begin{align}
  & \Rightarrow 13\left( \dfrac{17}{60} \right) \\
 & \therefore \operatorname{cosec}\theta +\sec \theta =\dfrac{221}{60} \\
\end{align}\]

Note:
It is important to note that while calculating the values of the respective functions using the corresponding trigonometric identities and functions we need to check the values of the numerator and denominator that are to be formed. Because interchanging any of the numerator and denominator terms in any of the equations completely changes the result.
Instead of using the above trigonometric identities to independently find the values of the secant and cosecant functions we can also find them directly from the other trigonometric identities which directly gives the cosecant and secant values.
\[\begin{align}
  & {{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1 \\
 & {{\operatorname{cosec}}^{2}}\theta -{{\cot }^{2}}\theta =1 \\
\end{align}\]
Both the methods give the same value.