Question

How do you solve $ \sec x\csc x=2\csc x $ and find all solutions in the interval $ [0,2\pi ) $ ?

Answer 1

Hint: We can solve the above given trigonometric equation in the question by using certain transformations in the equation and on further simplifying and solving the equation we can get all the solutions in the given interval in the given question.

Complete step by step answer:
From the question we had been given that, $ \sec x\csc x=2\csc x $
Now, we have to do certainly suitable transformations to get the equation more simplified.
We know some basic trigonometric formula shown below, $ \csc x=\dfrac{1}{\sin x} $ and $ \sec x=\dfrac{1}{\cos x} $
By using the above formula we can simplify the given trigonometric equation. By applying the above formula we get the below equation, $ \sec x\csc x=2\csc x $
By substituting the above mentioned formula we will have $ \dfrac{1}{\cos x}.\dfrac{1}{\sin x}=2\dfrac{1}{\sin x} $
After cancellation of terms we will have $ \dfrac{1}{\cos x}=2 $ .
After simplifying we will have $ \cos x=\dfrac{1}{2} $
Now we have to find the intervals in which the above equation will be satisfied.
We know that, $ \cos \dfrac{\pi }{3}=\dfrac{1}{2} $
We also know that cosine is positive at first quadrant and fourth quadrant.
So for fourth quadrant the angle will be $ 2\pi -\theta $
 $ \cos x=\cos \left( 2\pi -\theta \right) $
 $ \cos x=\cos \left( 2\pi -\dfrac{\pi }{3} \right) $
 $ \cos x=\cos \left( \dfrac{5\pi }{3} \right) $
Therefore, solutions in the interval $ [0,2\pi ) $ are $ \dfrac{\pi }{3},\dfrac{5\pi }{3} $

Note:
 We should be well aware of the trigonometric basic formula. We should be also well aware of the values of the angles in which quadrant they are positive and in which quadrant they are negative. This question can be answered in a few steps simply like $ \begin{align}
  & \sec x\csc x=2\csc x \\
 & \Rightarrow \sec x=2 \\
 & \Rightarrow \cos x=\dfrac{1}{2} \\
\end{align} $
In the given interval $ [0,2\pi ) $ we will have two solutions $ \dfrac{\pi }{3} $ and $ \dfrac{5\pi }{3} $ .