Question

How do you simplify \[\dfrac{\cot \left( \theta \right)}{\cos \left( \theta \right)}\] ?

Answer 1

Hint: From the question given, we have been asked to simplify\[\dfrac{\cot \left( \theta \right)}{\cos \left( \theta \right)}\].
To solve the given problem, We have to use the basic formulae of trigonometry. After using the basic formulae of \[\cot \theta \] in trigonometry to the given question, we have to simplify further to get the final accurate and exact answer. We use the \[\text{cosec}\theta \] formula at the end to make the answer much more simplified.

Complete step by step solution:
From the question, we have been given that,
\[\Rightarrow \dfrac{\cot \left( \theta \right)}{\cos \left( \theta \right)}\]
From the basic formulae of trigonometry, we already know that,
\[\Rightarrow \cot \theta =\dfrac{\cos \theta }{\sin \theta }\]
Now, we have to substitute the above formula in the given question.
By substituting the above formula in the given question, we get
\[\Rightarrow \dfrac{\dfrac{\cos \left( \theta \right)}{\sin \left( \theta \right)}}{\cos \left( \theta \right)}\]
Now, as we have already discussed earlier, we have to simplify further to get the exact answer for the given question.
By simplifying the above obtained trigonometric expression further, we get
\[\Rightarrow \dfrac{\cot \left( \theta \right)}{\cos \left( \theta \right)}=\dfrac{\cos \left( \theta \right)}{\sin \left( \theta \right)\times \cos \left( \theta \right)}\]
Now, \[\cos \left( \theta \right)\] will be cancelled in both the numerator and denominator.
By elimination of \[\cos \left( \theta \right)\] the final trigonometric expression we will get is
\[\Rightarrow \dfrac{\cot \left( \theta \right)}{\cos \left( \theta \right)}=\dfrac{1}{\sin \left( \theta \right)}\]
Now, we know the trigonometric relations between sine and cosecant trigonometric expression.
We know that
\[\Rightarrow \dfrac{1}{\sin \left( \theta \right)}=\text{cosec}\left( \theta \right)\]
Therefore,
\[\Rightarrow \dfrac{\cot \left( \theta \right)}{\cos \left( \theta \right)}=\text{cosec}\left( \theta \right)\]
Hence, the given question is simplified by using the basic formulae of trigonometry and general identity of trigonometry.

Note: Students should be well aware of the basic formulae of trigonometry. Students should know the general relations between the \[\sin \theta \] and \[\cos \theta \], \[\cot \theta \] and \[\tan \theta \] functions in trigonometry. The relations between them are
\[\Rightarrow \dfrac{1}{\sin \left( \theta \right)}=\text{cosec}\left( \theta \right)\]
\[\Rightarrow \dfrac{1}{\cos \left( \theta \right)}=sec\left( \theta \right)\]
\[\Rightarrow \dfrac{1}{\tan \left( \theta \right)}=\cot \left( \theta \right)\]
These are the basic relations of trigonometry and these conversions are very useful in simplifying the trigonometric expressions. students should be very careful while conversion of trigonometric expressions and also while doing the calculation part.