Question

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

Answer 1

Hint: In this question first of all we will consider the left hand side of the given equation and using different identities of trigonometric functions try to get the right hand side of the equation as its final answer. For that first of all we will convert all the left hand side terms into sine and cosine functions because we want our answer as \[\cos ecx\].
Therefore after converting the left hand side functions into sine and cosine terms simplify it further using standard identities of trigonometric function.

Complete solution:
We have given
\[\sin x+\cos x.\cot x=\csc x\]
Let us consider,
Left hand side $L.H.S.=\sin x+\cos x.\cot x$
AS we know that \[\cot x=\dfrac{\cos x}{\sin x}\]. So substitute it in the above expression in place of \[\cot x\].
Left hand side $L.H.S.=\sin x+\cos x.\left( \dfrac{\cos x}{\sin x} \right)$
$=\sin x+\dfrac{{{\cos }^{2}}x}{\sin x}$
$=\dfrac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\sin x}$
As we know the trigonometric identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$
$\therefore L.H.S.=\dfrac{1}{\sin x}$
As we know that reciprocal of \[\sin x\] i.e. \[\dfrac{1}{\sin x}\] is \[\csc x\].
\[\therefore L.H.S.=cosecx\]
\[\Rightarrow L.H.S.=R.H.S.\]

Hence, \[\sin x+\cos x\cot x=\cos ecx\] is proved.

Note:
In order to solve this type of question these are some general rules to remember which are mentioned below:
Some trigonometry equations contain more than one trigonometric function. Others have mixtures of multiple angles and single angles with the same variable or sometimes we have two or more different trigonometric functions. Therefore whenever we have equations with multiple angles or single angles we will try to convert it into more manageable form so that we can use factoring or another method to solve them, we can use identities to substitute for some or all of the terms.
If we have an equation made up of two or more trigonometric functions a good tactic is to replace each function by using either a ratio identity or a reciprocal identity.
Using these identities creates fractions and fractions require a common denominator, and by the way having a fraction in trigonometric equations is good because it helps to substitute some standard identities which will help to make expression much simpler.
Some important identities:
(1) Pythagorean Identities:
$\Rightarrow {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
$\Rightarrow {{\sec }^{2}}\theta =1+{{\tan }^{2}}\theta $
$\Rightarrow \text{cose}{{\text{c}}^{\text{2}}}\theta =1+{{\cot }^{2}}\theta $
(2) Reciprocal identities:
$\Rightarrow $$\sec \theta =\dfrac{1}{\cos \theta }$
$\Rightarrow \text{cosec}\theta \text{=}\dfrac{1}{\sin \theta }$
$\Rightarrow $$\cot \theta =\dfrac{1}{\tan \theta }$