Prove that:
$\left( {1 + {{\cot }^2}\theta } \right).\left( {1 - \cos \theta } \right).\left( {1 + \cos \theta } \right) = 1$
Last updated date: 26th Mar 2023
•
Total views: 310.2k
•
Views today: 8.87k
Answer
310.2k+ views
Hint- Use the following formulae ${\text{ }}\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }},{\text{ }}\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$
We have to prove
$\left( {1 + {{\cot }^2}\theta } \right).\left( {1 - \cos \theta } \right).\left( {1 + \cos \theta } \right) = 1$
Consider L.H.S
$\left( {1 + {{\cot }^2}\theta } \right).\left( {1 - \cos \theta } \right).\left( {1 + \cos \theta } \right)$
As we know
${\text{ }}\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }},{\text{ }}\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$
So, apply these trigonometric properties in above equation
$
\Rightarrow \left( {1 + {{\left( {\dfrac{{\cos \theta }}{{\sin \theta }}} \right)}^2}} \right)\left( {1 - {{\cos }^2}\theta } \right) \\
\Rightarrow \left( {\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta }}} \right)\left( {1 - {{\cos }^2}\theta } \right) \\
$
Now we know that ${\sin ^2}\theta + {\cos ^2}\theta = 1,{\text{ }}\left( {1 - {{\cos }^2}\theta } \right) = {\sin ^2}\theta $
So, apply these trigonometric properties in above equation
$
\Rightarrow \left( {\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta }}} \right)\left( {1 - {{\cos }^2}\theta } \right) = \dfrac{1}{{{{\sin }^2}\theta }}{\sin ^2}\theta \\
= 1 \\
$
= R.H.S
Hence Proved
Note- In such types of questions always remember the general trigonometric identities which are stated above and using these properties simplify the given equation we will get the required answer.
We have to prove
$\left( {1 + {{\cot }^2}\theta } \right).\left( {1 - \cos \theta } \right).\left( {1 + \cos \theta } \right) = 1$
Consider L.H.S
$\left( {1 + {{\cot }^2}\theta } \right).\left( {1 - \cos \theta } \right).\left( {1 + \cos \theta } \right)$
As we know
${\text{ }}\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }},{\text{ }}\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$
So, apply these trigonometric properties in above equation
$
\Rightarrow \left( {1 + {{\left( {\dfrac{{\cos \theta }}{{\sin \theta }}} \right)}^2}} \right)\left( {1 - {{\cos }^2}\theta } \right) \\
\Rightarrow \left( {\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta }}} \right)\left( {1 - {{\cos }^2}\theta } \right) \\
$
Now we know that ${\sin ^2}\theta + {\cos ^2}\theta = 1,{\text{ }}\left( {1 - {{\cos }^2}\theta } \right) = {\sin ^2}\theta $
So, apply these trigonometric properties in above equation
$
\Rightarrow \left( {\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta }}} \right)\left( {1 - {{\cos }^2}\theta } \right) = \dfrac{1}{{{{\sin }^2}\theta }}{\sin ^2}\theta \\
= 1 \\
$
= R.H.S
Hence Proved
Note- In such types of questions always remember the general trigonometric identities which are stated above and using these properties simplify the given equation we will get the required answer.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
