Question

Find the value of \[\sec \left( 90-\theta \right)-\operatorname{cosec}\theta \]?

Answer 1

Hint: We start solving the problem by equating the given equation to x. We then make use of the results $\sec \alpha =\dfrac{1}{\cos \alpha }$ and $\operatorname{cosec}\alpha =\dfrac{1}{\sin \alpha }$ to proceed through the problem. We then make use of the result $\cos \left( 90-\theta \right)=\sin \theta $ to proceed further through the problem. We then make use of the result $\dfrac{1}{\sin \theta }=\text{cosec}\theta $ and then make the necessary calculations to get the required answer.

Complete step by step answer:
According to the problem, we are asked to find the value of \[\sec \left( 90-\theta \right)-\operatorname{cosec}\theta \].
Let us assume $x=\sec \left( 90-\theta \right)-\operatorname{cosec}\theta $ ---(1).
We can see that the value of $\theta $ cannot be equal to $n\pi $, $\left( n\in Z \right)$ as we can see that $\operatorname{cosec}n\pi =\dfrac{1}{\sin n\pi }=\dfrac{1}{0}$ which is undefined and also $\sec \left( 90-n\pi \right)=\dfrac{1}{\cos \left( 90-n\pi \right)}=\dfrac{1}{\sin n\pi }=\dfrac{1}{0}$ which is also undefined.
We know that $\sec \alpha =\dfrac{1}{\cos \alpha }$ and $\operatorname{cosec}\alpha =\dfrac{1}{\sin \alpha }$. Let us use this results in equation (1).
\[\Rightarrow x=\dfrac{1}{\cos \left( 90-\theta \right)}-\dfrac{1}{\sin \theta }\] ---(2).
We know that $\cos \left( 90-\theta \right)=\sin \theta $. Let us use this result in equation (2).
\[\Rightarrow x=\dfrac{1}{\sin \theta }-\dfrac{1}{\sin \theta }\] ---(3).
We know that $\dfrac{1}{\sin \theta }=\text{cosec}\theta $. Let us use this result in equation (3).
\[\Rightarrow x=\text{cosec}\theta -\text{cosec}\theta \].
\[\Rightarrow x=0\], for $\theta \ne n\pi $.
So, we have found the value of x as 0.

$\therefore $ The value of \[\sec \left( 90-\theta \right)-\operatorname{cosec}\theta \] is 0 for $\theta \ne n\pi $.

Note: Whenever we get this type of problem, we first try to find the values of the independent variable at which the given function is not valid which makes a huge difference in the required solution. We can also make use of the result $\sec \left( 90-\theta \right)=\operatorname{cosec}\theta $ to solve the given problem. We should not make calculation mistakes while solving this type of problem. Similarly, we can expect problems to find the value of $\tan \left( 90-\theta \right)-\cot \theta $.