Question

If $0<\theta <90{}^\circ $ and $\sec \theta =\csc 60{}^\circ $, find the value of \[2{{\cos }^{2}}\theta -1\]?

Answer 1

Hint: By using the equation given in the question we will transform the value of $\theta $ in one trigonometric variable. After that, we will evaluate $\theta $. Since, we have got a definite value of $\theta $ in degrees we will put that particular value in the final expression to obtain the answer.

Complete step-by-step answer:
Mathematics includes the study of topics which are related to quantity, structure, space and change. Mathematics is related to all the phenomena occurring in the world. When mathematical structures are good models of real phenomena mathematical reasoning can be used to provide insight or predictions about nature. In mathematics, trigonometry plays a key role in evaluation of various quantities associated with triangles.
As per given in the question, let the first equation be $\sec \theta =\csc 60{}^\circ \ldots (1)$
Now, transforming equation (1) in single trigonometric variable by converting cosec to sec we get,
$\begin{align}
  & \csc \theta =\sec (90{}^\circ -\theta ) \\
 & \therefore \csc 60{}^\circ =\sec (90{}^\circ -60{}^\circ ) \\
 & \Rightarrow \sec 30{}^\circ \\
\end{align}$
Now, taking the inverse sec transformation we get,
$\begin{align}
  & {{\sec }^{-1}}(\sec \theta )={{\sec }^{-1}}(\sec 30{}^\circ ) \\
 & \theta =30{}^\circ \\
\end{align}$
So, the value of $\theta $ is 30 in degrees.
Now, let another equation be $2{{\cos }^{2}}\theta -1\ldots (2)$.
Now, putting the value of $\theta $ as 30 degrees in equation (2) we get, $2{{\cos }^{2}}30{}^\circ -1$.
From the trigonometric table, the value of cos 30 is $\dfrac{\sqrt{3}}{2}$.
$\begin{align}
  & \Rightarrow 2\times {{\left( \dfrac{\sqrt{3}}{2} \right)}^{2}}-1 \\
 & \Rightarrow 2\times \dfrac{3}{4}-1 \\
 & \Rightarrow \dfrac{3}{2}-1 \\
 & \Rightarrow \dfrac{1}{2} \\
\end{align}$
So, the final answer is 0.5.

Note: This question can be alternatively solved by using the identity of $\cos 2\theta $ as the final expression which is given in the question is equivalent to $\cos 2\theta $. So, after evaluating theatres 30° we can directly put it in $\cos 2\theta $ and obtain the answer.