Question

Prove the following statement:
\[\dfrac{\operatorname{cosec}A}{\cot A+\tan A}=\cos A\]

Answer 1

Hint: First of all, consider the LHS of the given equation and convert the whole expression in terms of \[\sin \theta \] and \[\cos \theta \] by using the formulas \[\operatorname{cosec}\theta =\dfrac{1}{\sin \theta },\tan \theta =\dfrac{1}{\cot \theta }=\dfrac{\sin \theta }{\cos \theta }\]. Now, simplify the given expression to prove the desired result.
Complete step-by-step answer:
In this question, we have to prove that \[\dfrac{\operatorname{cosec}A}{\cot A+\tan A}=\cos A\]. Let us consider the LHS of the equation given in the question.
\[E=\dfrac{\operatorname{cosec}A}{\cot A+\tan A}\]
We know that \[\operatorname{cosec}\theta =\dfrac{1}{\sin \theta }\]. By using this in the numerator of the above expression, we get,
\[E=\dfrac{\dfrac{1}{\sin A}}{\cot A+\tan A}\]
We know that \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] and \[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\]. By using these in the denominator of the above expression, we get,
\[E=\dfrac{\dfrac{1}{\sin A}}{\dfrac{\cos A}{\sin A}+\dfrac{\sin A}{\cos A}}\]
By taking \[\sin A\cos A\] as LCM in the denominator and simplifying the expression, we get,
\[E=\dfrac{\dfrac{1}{\sin A}}{\dfrac{\cos A.\cos A+\sin A.\sin A}{\sin A\cos A}}\]
\[E=\dfrac{\dfrac{1}{\sin A}}{\dfrac{{{\cos }^{2}}A+{{\sin }^{2}}A}{\sin A\cos A}}\]
We know that \[{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1\]. By using this in the above expression, we get,
\[E=\dfrac{\dfrac{1}{\sin A}}{\dfrac{1}{\sin A\cos A}}\]
By simplifying the above expression, we get,
\[E=\dfrac{1}{\sin A}.\dfrac{\sin A\cos A}{1}\]
By canceling the like terms in the above expression, we get,
\[E=\cos A\]
E = RHS
So, we get, LHS = RHS
Hence proved.
So, we have proved that \[\dfrac{\operatorname{cosec}A}{\cot A+\tan A}=\cos A\].

Note: In these types of questions, it is always better to convert the whole equation in terms of \[\sin \theta \] and \[\cos \theta \] and then simplify it to get the desired result. Also, students often make mistakes while taking the LCM in questions related to trigonometry. So, this must be taken care of. Finally, students must remember the general trigonometric formulas to solve the question easily.