Question

How do you prove $\csc x - \sin x = \cos x\cot x$ ?

Answer 1

Hint:
Whenever we are given with the trigonometric function we need to know how a function can be represented in terms of other functions so that we can simplify and prove the given function. In the given function $\csc x - \sin x = \cos x\cot x$ first consider the left hand side function and replace the $\csc x$ by $\dfrac{1}{{\sin x}}$ and simplify the expression which gives the right hand side function.

Complete step by step solution:
Here in this question they have given the trigonometric functions to prove. Whenever we are given with the trigonometric function we need to know how a function can be represented in terms of other functions so that we can simplify and prove the given function.
The given function is $\csc x - \sin x = \cos x\cot x$ , in order to prove the given function consider the left hand side function try to simplify to get the right hand side function.
Therefore, the left hand side function is,
$L.H.S. = \csc x - \sin x$
Now, for simplification purposes, we can replace the function $\csc x$ by $\dfrac{1}{{\sin x}}$ which is by the reciprocal identity of the trigonometric function.
$ \Rightarrow L.H.S. = \csc x - \sin x = \dfrac{1}{{\sin x}} - \sin x$
We can rewrite the above expression as
$ \Rightarrow L.H.S. = \dfrac{1}{{\sin x}} + \dfrac{{ - \sin x}}{1}$
On simplifying the above expression, we get
$ \Rightarrow L.H.S. = \dfrac{{1 - (\sin x)(\sin x)}}{{\sin x}}$
$ \Rightarrow L.H.S. = \dfrac{{1 - {{\sin }^2}x}}{{\sin x}}$
From the trigonometric identities we have ${\cos ^2}x = 1 - {\sin ^2}x$ , by using this in above expression, we get
$ \Rightarrow L.H.S. = \dfrac{{{{\cos }^2}x}}{{\sin x}}$
The above expression can be written as
$ \Rightarrow L.H.S. = \cos x\dfrac{{\cos x}}{{\sin x}}$
We know that $\dfrac{{\cos x}}{{\sin x}} = \cot x$ , so by substituting this in the above expression, we get
$ \Rightarrow L.H.S. = \cos x\cot x = R.H.S.$

Hence $\csc x - \sin x = \cos x\cot x$ proved.

Note:
Whenever we have questions related to trigonometric function you need to remember all the standard trigonometric formulas which we can use for simplification purposes. This is just based on the substitutions, we can only solve the problem if we know the formulas.