Question

How do you simplify \[\cot \theta \sec \theta \sin \theta \]?

Answer 1

Hint: We have been given with trigonometric functions which has to be simplified. We will be using trigonometric formulae to solve them. Firstly, we will take them separately and write them in terms of sine and cosine functions and then combine them together and then we will simplify it further to obtain the simplified form.

Complete step by step solution:
According to the given question, we have trigonometric functions which we have to simplify. And we will be using trigonometric formulae to simplify the expression. We will be taking the three functions, in the given expression and simplify them. Later, we will combine them and cancel out the similar terms to get the simplified form.
We will start by writing the given expression, we have,
\[\cot \theta \sec \theta \sin \theta \]-----(1)
We will first take \[\cot \theta \], writing in terms of sine and cosine function, we get,
\[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\]-----(2)
As we know that tangent function and cotangent function are inverse of each other, that is, \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] then, \[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\].
We will now take \[\sec \theta \], simplifying it , we get,
\[\sec \theta =\dfrac{1}{\cos \theta }\]----(3)
Substituting equations (2) and (3) in equation (1), we get,
\[\cot \theta \sec \theta \sin \theta \]
\[\Rightarrow \dfrac{\cos \theta }{\sin \theta }\times \dfrac{1}{\cos \theta }\times \sin \theta \]
We have \[\cos \theta \] both in the numerator as well as in the denominator, so \[\cos \theta \] gets cancelled. And we get,
\[\Rightarrow \dfrac{1}{\sin \theta }\times \sin \theta \]
Similarly, we have \[\sin \theta \] both in the numerator and in the denominator as well, so \[\sin \theta \] gets cancelled as well. And we get,
\[\Rightarrow 1\]

Therefore, the simplified form of \[\cot \theta \sec \theta \sin \theta =1\].

Note: The simplification of the individual trigonometric function should be done carefully and the relation between the different functions must be remembered to prevent any errors. For example – while writing the simplification for \[\cot \theta \], then the relation between the tangent function and the cotangent function must be remembered and we know that they are inverse of each other. So, if the simplification of \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\], then \[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\] as we know \[\cot \theta =\dfrac{1}{\tan \theta }\].