Question

How to verify $\dfrac{{{\csc }^{3}}x-\csc x{{\cot }^{2}}x}{\csc x}=1$ ?

Answer 1

In order to verify the relation, we firstly simplify the equation by converting cosecant and cotangent functions to the sine and cosine functions form by using the conversion formulae, then by using the identities for the trigonometric functions, the question can be solved.

Formula used:
The formulae used in this question are the trigonometric relations.Firstly, to convert cosecant and cotangent, the formulae used are

\Rightarrow\cot x=\dfrac{\cos x}{\sin x} \\\ $$ And the relation, $$1-{{\cos }^{2}}x={{\sin }^{2}}x$$ So, these are the formulae used for solving the question. **Complete step by step solution:** In order to solve the question we start by simplifying the relation. So, let us first write the equation that we need to verify, $$\dfrac{{{\csc }^{3}}x-\csc x{{\cot }^{2}}x}{\csc x}=1$$ Now let us take the left and the right hand sides of the equations, and prove them to be equal. For this, we need to simplify the left hand side. This can be done by converting the cosine and the cotangent functions into the sine and cosine forms by using the relations: $$\csc x=\dfrac{1}{\sin x} \\\ \Rightarrow\cot x=\dfrac{\cos x}{\sin x} \\\ $$ Now simplifying the relation, $$\dfrac{\dfrac{1}{{{\sin }^{3}}x}-\left( \dfrac{1}{\sin x} \right)\dfrac{{{\cos }^{2}}x}{{{\sin }^{2}}x}}{\dfrac{1}{\sin x}} \\\ \Rightarrow \dfrac{\dfrac{1}{{{\sin }^{3}}x}-\dfrac{{{\cos }^{2}}x}{{{\sin }^{3}}x}}{\dfrac{1}{\sin x}} \\\ \Rightarrow \dfrac{\dfrac{1-{{\cos }^{2}}x}{{{\sin }^{3}}x}}{\dfrac{1}{\sin x}} \\\ $$ Using the relation, $$1-{{\cos }^{2}}x={{\sin }^{2}}x$$ $$\dfrac{\dfrac{{{\sin }^{2}}x}{{{\sin }^{3}}x}}{\dfrac{1}{\sin x}} \\\ \Rightarrow \dfrac{\dfrac{1}{\sin x}}{\dfrac{1}{\sin x}} \\\ \Rightarrow 1 \\\ $$ Hence, $$L.H.S=1$$ Also, we know that, $$R.H.S=1$$ **As, $$L.H.S=R.H.S=1$$. Hence, the equation is verified.** **Note:** It is a very important step to simplify the functions given in the question to the sine and the cosine forms, it makes it really easy to solve the question, however, more direct formulae for the cosecant and the cotangent functions can also be applied but that makes it quite lengthy and complicated. So, it is always better to convert to sine and cosine forms.